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SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 



This document contains information affecting the national defense of the 
United States within the meaning of the Espionage Act, 50 U. S. C., 31 and 
32, as amended. Its transmission or the revelation of its contents in any 
manner to an unauthorized person is prohibited by law. 

This volume is classified CONFIDENTIAL in accordance with security 
regulations of the War and Navy Departments because certain chapters 
contain material which was CONFIDENTIAL at the date of printing. 
Other chapters may have had a lower classification or none. The reader is 
advised to consult the War and Navy agencies listed on the reverse of this 
page for the current classification of any material. 



Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Group of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol- 
ume was printed and bjound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has been 
made by the War and Navy Departments. Inquiries concerning the 
availability and distribution of the Summary Technical Report 
volumes and microfilmed and other reference material should be 
addressed to the War Department Library, Room lA-522, The 
Pentagon, Washington 25, D. C., or to the Office of Naval Re- 
search, Navy Department, Attention: Reports and Documents 
Section, Washington 25, D. C. 


Copy No. 

5 

This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to : 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATa) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 


REG«AD':D UNCUSSiF'.EO 


1310013 


SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC 


VOLUME 2B 


A THEORETICAL BASIS 
FOR METHODS OF SEARCH 
AND SCREENING 


Return To 

SCIENCE AND lECKNOlOfiy DIVISION 

Library of Congress 

OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 6 
JOHN T. TATE, CHIEF 


WASHINGTON, D. C., 1946 


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NATIONAL DEFENSE RESEARCH COxMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 


a -JAN12 
_«H|( Mtn tfi 


Roger Adams Army Representative^ 

Frank B. Jewett Navy Representative^ 

Karl T. Compton Commissioner of Patents^ 

Irvin Stewart, Executive Secretary 


^Army representatives in order of service: 


Maj. Gen. G. V. Strong 
Maj. Gen. R. C. Moore 
Maj. Gen. C. C. Williams 
Brig. Gen. W. A. Wood, Jr. 

Col. E. 


Col. L. A. Denson 
Col. P. R. Faymonville 
Brig. Gen. E. A. Regnier 
Col. M. M. Irvine 
. Routheau 


"^Navy representatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Purer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore H. A. Schade 
^Commissioners of Patents in order of service: 

Conway P. Coe Casper W. Ooms 


- NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities of 
warfare, together -with contract facilities for carrying out 
these projects and programs, and (2) to administer the tech- 
nical and scientific work of the contracts. More specifically, 
NDRC functioned by initiating research projects on requests 
from the Army or the Navy, or on requests from an allied 
government transmitted through the Liaison Office of OSRD, 
or on its own considered initiative as a result of the experi- 
ence of its members. Proposals prepared by the Division, 
Panel, or Committee for research contracts for performance 
of the work involved in such projects were first reviewed by 
NDRC, and if approved, recommended to the Director of 
OSRD. Upon approval of a proposal by the Director, a con- 
tract permitting maximum flexibility of scientific effort was 
arranged. The business aspects of the contract, including 
such matters as materials, clearances, vouchers, patents, 
priorities, legal matters, and administration of patent matters 
were handled by the Executive Secretary of OSRD. 

Originally NDRC administered its work through five 
divisions, each [headed by one of the NDRC members. 
These were: 

Division A — Armor and Ordnance 

Division B — Bombs, Fuels, Gases, & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three ad- 
ministrative divisions, panels, or committees were created, 
each with a chief selected on the basis of his outstanding 
work in the particular field. The NDRC members then be- 
came a reviewing and advisory group to the Director of 
OSRD. The final organization was as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 


iv 



' nfiGRADSD ,U.^'3aASSIS'iiD 
OkltiK SiC ei US X 6 1 ■ ^ 


I V'rary of Congress 



201 5 490945 


NDRC FOREWORD 


AS EVENTS of the years preceding 1940 revealed 
jiY niore and more clearly the seriousness of the 
world situation, many scientists in this country came 
to realize the need of organizing scientific research for 
service in a national emergency. Recommendations 
which they made to the White House were given care- 
ful and sympathetic attention, and as a result the 
National Defense Research Committee [NDRC] was 
formed by Executive Order of the President in the 
summer of 1940. The members of NDRC, appointed 
by the President, were instructed to supplement the 
work of the Army and the Navy in the development 
of the instrumentalities of war. A year later, upon the 
establishment of the Office of Scientific Research and 
Development [OSRD], NDRC became one of its 
units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to sum- 
marize and evaluate its work and to present it in 
a useful and permanent form. It comprises some 
seventy volumes broken into groups corresponding to 
the NDRC Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s report 
contains a summary of the report, stating the prob- 
lems presented and the philosophy of attacking them 
and summarizing the results of the research, develop- 
ment, and training activities undertaken. Some vol- 
umes may be “state of the art” treatises covering 
subjects to which various research groups have con- 
tributed information. Others may contain descrip- 
tions of devices developed in the laboratories. A 
master index of all these divisional, panel, and com- 
mittee reports which together constitute the Sum- 
mary Technical Report of NDRC is contained in a 
separate volume, which also includes the index of a 
microfilm record of pertinent technical laboratory re- 
ports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of suf- 
ficient popular interest that it was found desirable to 
report them in the form of monographs, such as the 
series on radar by Division 14 and the monograph on 
sampling inspection by the Applied Mathematics 


Panel. Since the material treated in them is not 
duplicated in the Summary Technical Report of 
NDRC, the monographs are an important part of 
the story of these aspects of NDRC research. 

In contrast to the information on radar, which is 
of widespread interest and much of which is released 
to the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of Di- 
vision 6 is found almost entirely in its Summary Tech- 
nical Report, which runs to over twenty volumes. The 
extent of the work of a Division cannot therefore be 
judged solely by the number of volumes devoted to it 
in the Summary Technical Report of NDRC : account 
must be taken of the monographs and available re- 
ports published elsewhere. 

Any great cooperative endeavor must stand or fall 
with the will and integrity of the men engaged in it. 
This fact held true for NDRC from its inception, and 
for Division 6 under the leadership of Dr. John T. 
Tate. To Dr. Tate and the men who worked with 
him — some as members of Division 6, some as repre- 
sentatives of the Division’s contractors — belongs the 
sincere gratitude of the Nation for a difficult and 
often dangerous job well done. Their efforts con- 
tributed significantly to the outcome of our naval 
operations during the war and richly deserved the 
warm response they received from the Navy. In addi- 
tion, their contributions to the knowledge of the 
ocean and to the art of oceanographic research will 
assuredly speed peacetime investigations in this field 
and bring rich benefits to all mankind. 

The Summary Technical Report of Division 6, pre- 
pared under the direction of the Division Chief and 
authorized by him for publication, not only presents 
the methods and results of widely varied research and 
development programs but is essentially a record of 
the unstinted loyal cooperation of able men linked in 
a common effort to contribute to the defense of their 
Nation. To them all we extend our deep appreciation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. CoNANT, Chairman 
National Defense Research Committee 




'^2 1 0 0 1 3 


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FOREWORD 


T his report, constituting Volume 2B of the Divi- 
sion 6 Summary Technical Report, embodies 
some of the statistical and analytical work done dur- 
ing the period 1942-1945 by members of the Opera- 
tions Evaluation Group [OEG] in the Headquarters 
of the Commander-in-Chief, U. S. Nayy. This group, 
composed of civilian scientists, mathematicians, and 
statisticians, and initially designated the Anti-Subma- 
rine Warfare Operations Research Group [ASWORG], 
was established early in 1942 under the supervision , 
of the Atlantic Fleet antisubmarine warfare officer. 
Captain (later Rear Admiral) W. D. Baker. 

When Captain Baker requested the formation of 
the Group, he stated that the antisubmarine warfare 
unit, as then constituted, was not in a position to 
evaluate properly (1) the probable effectiveness of 
suggested new weapons and (2) developments and 
procedures based upon mathematical studies and 
other available records. 

Captain Baker defined what was desired and in- 
tended by quoting from a RAF Coastal Command 
memorandum on the subject, as follows : 

“Experience over many parts of our war effort has 
shown that such analysis can be of the utmost value, 
and the lack of such analysis can be disastrous. Prob- 
ably the main reason why this is so, is that very many 
war operations involve considerations with which 
scientists are specially trained to compete and in 
which serving officers are in general not trained. This 
is especially the case with all those aspects of opera- 
tions into which probability considerations and the 
theory of errors enter. Serving officers of the highest 
caliber are necessarily employed in important execu- 
tive posts, and are therefore, not available for detailed 
analytic work.” 

As to the type of personnel required to staff these 
operations, Captain Baker quoted from the same 
memorandum : 

“A considerable fraction of the staff of an opera- 
tional research section should be of the very highest 
standing in science, and many of them should be 
drawn from those who have had experience at the 
Service Technical Establishments. Others should be 
chosen for analytic ability, e.g., gifted mathemati- 
cians, lawyers, chess players, etc. An ORS which con- 
tents itself with the routine production of statistical 
reports and narratives will be of very limited value. 
The atmosphere required is that of a first class pure 


scientific research institution, and the caliber of the 
personnel should match this. All members of an ORS 
should spend part of their time at operational stations 
in close touch with the flying personnel, and where 
possible, should occasionally go on operational or 
training flights.” ' ' 

That the project should be undertaken and ,the pro- 
posed group established in the Navy was never ques- 
tioned. The experience of the British and of large 
American industries in carrying on very similar op- 
erational analyses made certain that a suitably staffed 
group could be very effective. The project was as- 
signed to Section C-4, NDRC, and under its contract 
with Columbia University a staff was recruited. 

Professor P. M. Morse, of the Massachusetts In- 
stitute of Technology, became project supervisor, and 
Dr. William Shackley was granted a leave of absence 
from the Bell Telephone Laboratories to act as di- 
rector of research. ' 

The operations performed by ASWORG closely 
followed Captain Baker’s conception of how the 
group could assist the Navy. Also, Professor Morse 
was able to recruit personnel with qualifications very 
similar to those specified by Captain Baker. Although 
organizational changes occurred, the objectives of the 
Group itself were not altered. With the creation of 
the Tenth Fleet as a means of coordinating antisub- 
marine warfare, the group now under the sponsorhip 
of the Office of Field Service [OFS] was made an in- 
tegral part of the Fleet staff. As the activities of the 
group expanded to cover other fields, the name was 
shortened to Operations Research Group [ORG]. At 
the end of the war, the group was reorganized as the 
Operations Evaluation Group and assigned to the 
staff of the Chief of Naval Operations under a direct 
Navy contract with the Massachusetts Institute of 
Technology. It is by this latest organization that the 
present report has been prepared. 

The material summarized in this volume has been 
compiled from reports and memoranda issued by the 
group and from the first hand knowledge and experi- 
ence which the authors gained during the war with 
the techniques and problems discussed. Specific ex- 
amples are developed of the applications of the more 
general methods described in Volume 2 A, Methods 
of Operations Research, to the problems of submarine 
warfare. Also, a mathematical basis is provided for 
Volume 3, A Summary of A*SIF Operations in World 

vii 


3 


REGRADED LiNCLASSli^D 

5 xC ARE/ £ / TAS 2 1001 



FOREWORD 


Vlll 


War II, as well as for a wide category of similar in- 
vestigations. Although the tactical doctrines pre- 
sented apply to instruments, weapons, and conditions 
prevailing during this last war, it is believed that the 
methods and systematic processes of analysis which 
led to these doctrines have wide application — not 
only to submarine warfare but to many other mili- 
tary and civilian problems. 

The Division wishes to thank Professor Morse and 
his group as a whole for the work that underlies this 
volume, and especially to thank the principal author. 
Dr. B. O. Koopman. It is impossible to name the long 


regraded UNCLAS^BED 
ORDER SEC ARMY BY TAG 


list of Navy officers and some Army officers who gave 
support to the work of the Operations Research 
Group and who share credit for its performance. The 
full list would include many officers at operational 
bases and stations. However special acknowledg- 
ment should be made of the cooperation and assist- 
ance received from Admiral W. D. Baker, from Ad- 
miral F. S. Low, Chief of Staff, Tenth Fleet, of the 
U. S. Navy, and from the British operations research 
groups. 

John T. Tate 
Chief, Division 6 


PREFACE 


As THE Operations Research Group was at work 
1\. investigating one question after another in the 
course of its service to the Commander-in-Chief of 
the United States Navy, in World War II, it became 
progressively more apparent that large classes of 
problems were united by common bonds and could be 
handled by common methods, that there was indeed 
unity in diversity. And as in other fields of scientific 
endeavor, where the clarifying influence of general 
ideas and methods can form a body of isolated facts 
into a powerful theory — once they exist in sufficient 
number — so in the work of the Group, methods bor- 
rowed from the mathematician and mathematical 
physicist showed their power and usefulness in those 
classes of problems in which the body of practical in- 
formation had sufficiently accumulated. In this re- 
gard, one field was pre-eminently ripe for mathemati- 
cal treatment : the field involving problems of search. 

In every question of search there are in principle 
two parts. One involves the targets, and studies their 
physical characteristics, position, and motion; since 
from the very nature of the problem the latter are 
largely unknown to the searcher, a branch of the 
science of probability is applied, sometimes so simple 
as to be trivial, at other times involving developments 
comparable to statistical mechanics. The other part 
involves the searcher, his capabilities, position, and 
motion; inasmuch as detection is an event fraught 
with manifold uncertainties, this part of the question 
will also appeal to probability, specifically studying 
the probability laws of detection. But the study does 
not stop here : having gained fundamental knowledge 
as to these two parts of the question and their inter- 
relation, it is necessary to make application to the 
tactical matters in which search is an essential com- 
ponent, such as hunts, barriers, and those defensive 
types of search known as screens. 

The book treats these questions from the point of 
view and in the order indicated above. It is intended 
to be scientific and critical in spirit and mathematical 
in method, and while the data upon which its theory 
rests are practical and experimental and the ultimate 


application of its conclusions is to naval warfare, the 
book itself is not a manual of practical information 
for naval officers. Rather it is intended to serve as a 
theoretical framework and foundation for more im- 
mediately practical studies and recommendations. In 
particular, it stands in this relation to Yolume 3 of the 
present series {A Summary of Antisubmarine Warfare 
Operations in World War II). On the other hand, its 
relation to Volume 2 A {Methods of Operations Re- 
search) is in furnishing systematically developed ex- 
amples, on the analytical side, of the possibilities of 
operations research foreshadowed in that volume. It 
is intended for a reader having an interest of a scien- 
tific order in the matters treated. While nothing be- 
yond undergraduate physics and mathematics (calcu- 
lus) is required, a willingness to follow theoretical 
reasoning of a sometimes rather involved nature is 
assumed. 

The work has been to such a degree the result of a 
majority of the Operations Research Group that to 
render adequate acknowledgment would almost be 
tantamount to giving the roster of the Group: re- 
quirements of brevity confine us to the names of those 
who have been directly involved in writing parts of 
the book. We wish to express our thanks to Dr. E. S. 
Lamar for the chapter on visual detection; to Mr. 
T. E. Phipps for the chapter on radar detection; to 
Mr. A. M. Thorndike for the chapter on sonar detec- 
tion and for a part of the chapter on sonar screens; to 
Dr. J. Steinhardt for important help in the chapter on 
radar detection and for material on barrier patrols 
and defense of a landing operation in Chapter 7 ; to 
Dr. J. M. Dobbie for material on square searches in 
the latter chapter; to Mr. Milton Lewis for material 
on sonar screens; to Mr. J. A. Neuendorffer for ma- 
terial on aerial escort. Finally, it is our great pleasure 
to vouchsafe our indebtedness to Dr. G. E. Kimball, 
the pioneer in the theory of search, without whose 
help and inspiration this enterprise might never have 
been undertaken. 

B. O. Koopman 


REGRADSD UNCLASSHRED 


1 AO 1 


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CONTENTS 


CHAPTER 

1 Position, Motion, and Random Encounters 

2 Target Detection 

3 The Distribution of Searching Effort 

4 Visual Detection 

5 Radar Detection 

6 Sonar Detection 

7 The Search for Targets in Transit . 

8 Sonar Screens 

9 Aerial Escort 

Glossary 

Contract Numbers 

Service Project Numbers 

Index 





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POSITION, MOTION, 


Chapter 1 

AND RANDOM ENCOUNTER 



11 INTRODUCTION 

Among the important functions of any naval 
1\. operation is the detection (and location) of the 
enemy. Detection has acquired the stature of a science, 
and it is to the foundations of this science of detec- 
tion that the present work is devoted. 

A first aspect of any problem of detection concerns 
the properties of the instrument of detection: the 
properties of the eye, the characteristics of the radar 
set, or the nature and capabilities of the sonar equip- 
ment, and similarly for any other mechanism of de- 
tection which it is proposed to employ. This aspect of 
the question, which forms the subject of Chapters 4, 
5, and 6, is called the contact problem. It is essentially 
a study of engineering or, in the case of the eye, of 
physiology, but its conclusions have to be given in 
terms of probability — the probability of detection. 

A second aspect of problems of detection concerns 
the path and motion of the searching unit (termed 
the observer) in its relation to the presumed position 
and motion of the object of search (the target, as it 
shall be called throughout). This presupposes that 
the contact problem has been solved to a satisfactory 
approximation and employs appropriate methods of 
geometry, relative motion, and probability. This as- 
pect of the question is called the track problem and 
forms the subject of Chapters 7, 8, and 9, where such 
matters as searches, barriers, hunts, and screens are 
studied. 

A third aspect — and this will pervade all our later 
chapters, but particularly Chapter 3 — is that of force 
requirements and their economy. As in other naval 
operations, effectiveness can be increased to perfec- 
tion if no limit is set to the forces at our disposal, but 
the realistic problem is to achieve the greatest effect 
with limited forces, or, equivalently, to achieve a re- 
quired effect with the greatest economy of forces. 

The scope of the book necessitates the omission of 
such matters as the use and deployment of striking 
forces in conjunction with detection. 

The present chapter provides the general defini- 
tions and geometrical and statistical methods which 
are required in all later parts of the work. Chapter 2 
deals with the generalities involved in detection. 


Viewed largely and in its ends, the science of de- 
tection is a branch of tactics. But like all other branches 
of tactics it achieves its ends by leading through 
engineering, physics, physiology, mathematics, and 
statistics. And it is an ever progressing science: 
While the main emphasis of this book is to study the 
existing knowledge and its application to tactical 
problems, sight is not lost of the inverse process, that 
of improving the theory by knowledge gained in its 
application, the study of operational data. 

12 MOTION AT FIXED SPEED AND COURSE 

In a most important case both observer and target 
are moving at constant speeds in straight lines: 

V = speed of observer in knots (ocean or true speed), 
u = speed of target in knots (ocean or true speed), 
w = speed of target relative to observer in knots. 

The relationship oi w io u and v is best shown by 
drawing the vector velocities u, v, w, whereupon it is 
seen that w is simply the vector difference w = u — v 
(Figure 1). For, w being the target’s velocity with 



Figure 1. True and relative velocities and angles. 


respect to a reference system, itself moving over the 
ocean at velocity v, the vector sum w + v must equal 
the target’s ocean velocity u; hence the above equa- 
tion. Figure 1 also shows two important angles, the 


^ OQP^PfW^NTIAL 


1 


2 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


target’s track angle 4> and relative course 6, with respect 
to the observer, where 

<f) = angle between v and u measured from the 
former to the latter in the clockwise sense. 

6 = angle between v and w measured from the 
former to the latter in the clockwise sense. 

Throughout the present chapter, and later unless the 
contrary is explicitly stated, these angles are meas- 
ured in radians and 0^<^<2 7r, O^0<2 7r. 

A convenient method for showing the dependence 
of the relative quantities w and 6 upon the angle 4> 
(the speeds u and v remaining fixed) is by drawing 
the circular diagrams A, B, and C of Figure 2, cor- 
responding to the cases v < u, v = u, and v > u, 
respectively. In each case the radius of the circle 
around which the extremities of u and w move is u, 
and the distance of its center from the origin 0 of w 
(the extremity of v) is v. As </> goes from 0 to 2 tt, v 
stays fixed, u rotates with its length remaining con- 
stant, and w changes both in length and direction. It 
is to be noted that while in case A (v < u) all direc- 
tions of w are possible (0 ^ 0 < 2 7r), in the other 
cases {v ^ u) this is untrue, and we have : 

When V = u, - = B ^ — ; 

2 2 

When V > u,T — sin“^ — ^ 6 = tt sin~^ - • 

V V 

This corresponds with the fact that when the searcher 
is faster than the target, relative approach of the 
latter to the former is restricted (see Section 1.3). 
When V > u, two values of w correspond to general 
values of 6 for which approach is possible, one for the 
target approaching the observer, the other for the 
overtaking of a target headed away from the ob- 
server. When 6 = T ± sin“^ u/ v, there is just one 
value of w; for other B’s, no value. 

The relative speed w can be found from the law of 
cosines or else by projecting v and u on w and using 
the law of sines; similarly for </>. This expresses rela- 
tive quantities in terms of true: 

w =\/u^ — 2uv cos 4>, 

= — V cos 6 + •>/ u^ — sin^ 6, (1) 

sm 6 = - sm 0. 
w 

In addition to the speeds and angles just consid- 
ered, it is necessary to have further quantities to 



v<.u 





Figure 2. Circle of relative velocities. 


MOTION AT FIXED SPEED AND COURSE 


3 


specify a particular contact between observer and 
target; one must be able to state the position of the 
target relative to the observer at any given instant of 
time (epoch) t. One method of accomplishing this is 
to give the target range r and relative bearing (3; these 
are shown in Figure 3 (at the arbitrary epoch t), to- 


TARGET’S TRACK 



Figure 3. Geographic tracks, ranges, and bearings. 


gether with the target angle a, which depends on 
quantities previously introduced. The definitions are 
as follows: 

r = vector from observer to target, 
r = length of r in miles, ^ 

P = angle from v to r measured clockwise, 
a = angle from u to — r measured clockwise. 

As usual, angles are in radians and lie between 0 and 
2 TT, except when in later chapters the contrary is ex- 
plicitly stated. It is evident that when the roles of 
target and searcher are interchanged, those of ^ and a 
are likewise, vector r being replaced by the reversed 
vector — r. 

The situation relative to the observer is given in 
Figure 4, which shows the target’s track, etc., in a 
plane in which the observer is fixed and which moves 
over the ocean with the velocity v. It is seen that 
(r,jd) are the polar coordinates of the target referred 
to observer’s position and heading. The target’s track 
is altogether different from his geographic track of 
Figure 3; it is described with the velocity w, but the 
target’s heading is in the direction of u and hence not 
along its relative track. It is seen that with the par- 
ticular angles of Figure 4, o: = tt + /3 — <^. It is 

a Throughout, a “mile” shall mean a “nautical mile.” In nu- 
merical examples, a nautical mile is taken as 2,000 yards. 


sometimes convenient to use rectangular coordinates 
{^,7}), the 7} axis being along the observer’s heading. 
^ and 7] are in miles; they are related to (r,/3) by the 
equations 

= ^2 _[_ ^ 2 ^ ^ = r sin /3, r; = r cos (3. (2) 



Figure 4. Target’s track relative to observer. 


In the course of time (as t increases) u, v, w, u, v, w, 
(f), 6 stay constant, while r, r, /? (and a) change. If at 
the epoch t = ^o, r, r, have the values ro, ro, ^o, 
their values at a general epoch t are found by noting 
that relative to the observer the target undergoes the 
vector displacement Xt — ^o)w, and thus 

r = To + (^ - ^o)w, (3) 

from which the equations expressing (r,/3) in terms of 
{rQ,^Q,t — to) are found by trigonometry, and similarly 
for a. Equation (3) or its equivalent in terms of (r,/3) 
are the equations of the target relative to the observer. 
In the very special case when target and observer 
have the same speed and direction, i. e., when u = v 
so that w = 0, equation (3) reduces to r = ro, corre- 
sponding with the fact that the target remains fixed 
relative to the observer. In all other cases, there is a 
least distance between the target and the observer. 
This distance is called the lateral range of the target. 

Let X = lateral range of the target in miles, and 
y — distance in miles traveled by the target 
relative to the observer since its closest 
approach (negative prior to closest ap- 
proach). 

If fo is now used to denote the epoch of closest ap- 
proach, evidently y = {t — to)w. Figure 5 shows the 




4 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


relation between x,y and the earlier quantities. 
Clearly and with the particular angles 

of Figure 5, ^ = 6 — cot^^ (2/A)* 

Thus it appears that, in addition to u, v, <i> (and their 
dependent w, B), in order to specify a particular con- 
tact we can use either (r,/3) (at a standard epoch) or 


I 



Figure 5. Lateral range of the target. 


ix,to) (or, indeed, any convenient independent func- 
tions of either pair); (r,j8) can be found where (x,to) 
are given, and vice versa, and either pair can be used 
as independent variables. On the other hand, only 
one quantity is needed to specify a type of contact, 


13 the region of approach 

In Figure 6A, A is a plane region fixed with respect 
to the observer; A, then, is moving straight ahead 
over the ocean surface with the velocity v. It may or 
may not be possible for a target capable of moving 
with the speed u and starting outside A to enter A. 
It is understood that the target is restricted to the 
speed u but can choose any direction, and has all the 
time it needs to try to enter A. Evidently if u > v, 
the target can always enter A ; but ii u ^ v this is no 
longer necessarily true, as for example when the tar- 
get starts behind A, In order to be able to enter A the 
target must have its starting point in a certain region 
B called the region of approach. Of course B is also 
attached to the observer and moves over the ocean 
with the velocity v. Figure 6A shows the construc- 
tion of B] when v points up the page, a line inclined 
at the angle sin“i u/v with the vector v to the right 
is drawn to the right of A and is moved toward A 
until it touches A; the part of the line above the 
lowest point of contact forms the right-hand bound- 
ary of B. Similarly for the left-hand boundary, the 
inclination being to the left and the contapt occurring 
on the left of A. The rear boundary is the forward 
boundary of A between the two rear points of con- 
tact. 

The justification of this construction is based on 
Figure 6B: In the time t a point starting at P and 




Figure 6. Construction of the region of approach. 


for example, x : Given u, v, </> and the range of closest 
approach x, the configuration (searcher and target 
and their tracks) is determined, but not the time at 
which the contact occurs. 


moving with the observer’s velocity will arrive at Q, 
PQ = vt; the circle centered at Q and of radius ut is 
the locus of positions from which the target must 
start if it is to close this point after the time t. By 


CGNI ' ^WI ' ^NTIAL 


RANDOM DISTRIBUTIONS OF TARGETS 


5 


considering all positive values of t and observing how 
the circle varies in both center and radius, it is seen 
that it will sweep out the whole angular region be- 
tween the two (fixed) tangents drawn from 0. Thus 
the whole upper angular space between the two tan- 
gents to the circle is the locus of all starting positions 
of the target if it is to close the moving point at any 
time after the latter leaves P. B is constructed by- 
letting P take on all positions in A, whereupon the 
upper angular space attached to P sweeps out B (in 
addition to A itself). 

When u = V, the tangents coalesce into a hori- 
zontal line tangent to A in the rear, and B is the 
upper half-plane above it (exclusive of A). 

An important naval application of the region of 
approach is to submarine warfare, where v is the speed 
of the convoy and A the region within which a tor- 
pedo must be fired to be in range of a ship of the 
convoy. What we have been terming the “target” 
may be thought of as a submerged submarine having 
the underwater speed u < v. Then evidently the sub- 
marine must be in B in order that it may be able to 
approach, submerged, to a torpedo firing position. 

The angle sin~^ u/v is called the limiting approach 
angle — the “limiting submerged approach angle” in 
the example of the submarine. 

Quite a different figure for B is obtained if the 
target is assumed to have a limited time {T hours) to 
make its approach to A. Then even when u> v ap- 
proach is not always possible. The construction of A 
in such a case is given in Figure 7A and B, when 


given all positions on A, the points of the attached 
circle cover B (thus B is bounded by the envelope of 
the circles). A similar construction is made when 
u < V, where the starting point region is that bounded 
by the two tangents as well as the larger intercepted 
arc of a circle disposed as in Figure 6B. 

An example of this second type of region A is the 
case in which account is taken of the limited endur- 
ance time of submerged run of the submarine in the 
previous example. 

14 RANDOM DISTRIBUTIONS OF TARGETS 

In the two preceding sections the motion of ob- 
server and target were given or precisely specified and 
the conclusions were exact. In Section 1.2 one ob- 
server and one target of stated speeds and tracks 
were assumed; in Section 1.3 the same was true for 
the observer and for the target^s speed, but a pre- 
cisely defined class of targets (those which enter A) 
was considered in defining B (the locus of their start- 
ing points) and the conclusion was the precise one : 
“The target can enter A if and only if it starts in B” 
Fundamentally different is the state of affairs in the 
present section, in which the notion of random is 
introduced and conclusions are stated in terms of 
probability. Instead of saying, “Under such and such 
conditions the target will necessarily do so and so,” 
we shall be saying, “Under such and such conditions 
the probability that the target will do so and so has 




Figure 7. The region of approach with hmited time. 


u > v; Figure 7B shows a circle of radius uT cen- 
tered at Q and vT units ahead of the starting point P; 
the circular region is the locus of starting positions 
from which the target can close the point. If P is 


this value,” or, equivalently, “This percentage of 
targets will on the average do so and so.” The im- 
portance of arriving at probabilities and statistical 
results in naval matters should be self-evident. 


6 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


To specify a target (always assumed in this chapter 
to be moving at constant speed and course) it is 
necessary to give its speed and heading (the vector u) 
and also its position at a particular epoch (i.e., when 
t = to). This requires in principle four independent 
quantities, such as u, 4>,r, By a random distribution 
of targets is meant either of the two following situa- 
tions : 

I. There are present a very large number of differ- 
ent targets, and what is known is not their velocities 
and positions, but the proportion or percentage which 
have the various possible velocities and positions. 

II. There is present only one target; its velocity 
and position are not precisely known, but the proba- 
bilities that it have the various possible velocities 
and positions are known. 

In these statements, the proportion or probability 
of targets ‘‘having such and such a velocity and posi- 
tion”^ must be interpreted to mean “having a veloc- 
ity and position within a stated closeness of such and 
such a velocity and position.” Thus if the above 
choice of quantities is made, it is the proportion or 
probability of targets having a speed between u and 
u -f du, track angle between and 4> + d(i>, range 
between r and r + dr, and bearing between ^ and 

-f d/3 which is in question; in many cases (but not 
all! see below) it can be represented to terms of first 
order in the differentials by p{u,<i),r ,^)dud4>drd^ , and 
the function p{u,4>,r,di) is the mean relative density (I) 
or probability density (II) of the distribution. Then 
the proportion or probability for a large class of veloci- 
ties and positions is obtained by integrating p(w,(/),r,/3) 
over all values of the class considered — a quadruple 
integration in the “space” of the “coordinates” 
(u,<^,r,/3). 

While the situations in I and II above appear to be 
quite different, they are in reality equivalent, or 
rather either one leads to the other. Thus from the 
very large number of targets in I we can think of an 
individual target chosen at random, all targets having 
the same chance of being chosen; this target will then 
be the single target to which the situation II applies. 
Reciprocally, if a very large number of targets is 
constituted from individual targets to each of which 
the state of affairs of II applies, the resulting swarm 
will be as described in I. The mean relative density 
(I) and the probability density (II) are equal. All 

bSuch probabilities might all be zero. With many distribu- 
tions the probability that the target be exactly at a pre-stated 
position is always zero. It is the probability that it lie in a 
pre-stated area which is of interest. 


this is a consequence of the law of large numbers in 
the theory of probability. Situation I is generally 
used to give a pictorial representation of II which 
might otherwise seem too abstract; but I has the dis- 
advantage of being rather unrealistic — if so many 
targets were actually all present they would be apt to 
interfere with one another physically. 

What we have termed a random distribution should 
more properly be called a known random distribution, 
to distinguish it from the case where the values of the 
probabilities are partly or wholly unknown: a fre- 
quent problem of importance in operational analysis 
is to find them by theoretical calculations or statisti- 
cal methods. 

One of the simplest and most important cases of 
random distribution of targets is that of the uniform 
distribution of targets of given speed u. It is one which 
complies with the three following requirements. 

a. The probability that the track angle <l> be be- 
tween (t>i and <^2 is proportional to <t )2 — </>! [and hence 
is equal to {<t >2 — 0i)/27r when 0i < (f) 2 ]. 

b. The probability that at any chosen epoch the 
target be in the area A (fixed in the ocean or else fixed 
relative to the observer; the two situations are here 
equivalent) is proportional to A (and hence, if the 
target is known to be in a larger area B containing A, 
the probability that it be in A is A/ B). 

c. The event of (j> being between 4>i and 02 on the 
one hand and the event of the target being in A on 
the other are independent events. If one of them is 
known to have occurred, the probability of the occur- 
rence of the other is the same as before. 

In the case of such a distribution, the probability 
density will (when the target is given to be in B) have 
the value p{4>,r,^) = r/2TrB', this is because rdrd^ is 
the element of area corresponding to a position of 
range and bearing between r and r A- dr and /5 and 
jS + d/3 respectively. It is to be noted that the proba- 
bility is p{<f>,r,(3)d(pdrdl3 and not p{u,4),r,^)dud4>drd^, 
as in the earlier example, i.e., neither u nor du occur; 
this is because the value of the target’s speed is sup- 
posed to be known. 

The case considered is of importance in naval oper- 
ations, since it corresponds to the situation in which 
the target is an enemy unit presumed to be running 
at the known speed of about u knots, but is in such a 
large area of ocean with so many possible intentions 
that nothing concerning its position or heading can 
be regarded as known. The chances that the observer 
will make various kinds of contacts with such a unit 
are studied in the following section. 




UNIFORMLY DISTRIBUTED TARGETS 


7 


It is noted that in the foregoing example the lan- 
guage of II is used. This is permissible in view of the 
equivalence of II with I, and there is manifestly no 
difficulty in rewording things to correspond to 1. 
Whichever of the two terminologies will be employed 
in the succeeding pages will be purely a matter of con- 
venience: this involves absolutely no inconsistency. 

15 random encounters with uni- 
formly DISTRIRUTED TARGETS 


of first order in the differential) 2RwNd4>/2'K. Hence 
the total number Nq is given by integration : 



RN / 

— I ^ — 2uv cos <f) d(i>y 

^ Jo 

(4) 

4:RN ^ 1^/2 

= {u-\- v) \ ^/l — sin^ a sin^ xf/ dxf/, 


When an observer is progressing on its course at 
the constant velocity v among a uniform random dis- 
tribution of targets of speed u, it is frequently impor- 
tant to know the proportions of targets which pass 
within the stated range of R miles of the observer. In 
some cases R may be the range within which the ob- 
server can sight the target (horizon distance); in 
others, the range within which the target can detect 
the observer’s presence; again, R may be effective 
gunfire range of observer against target, or vice versa. 
If a circle of radius R is pictured centered on the ob- 
server and moving along with it at velocity v over the 
ocean, the question becomes that of the proportion of 
targets entering the circle, or entering it at various 
specified bearings, or the chance that a target of 
given starting point shall enter the circle — a question 
of prohahility, in contrast with that of Section 1.3 
which was one of possibility. The three problems will 
be solved in turn. 

Problem 1. Let there be on the average N targets 
per square mile {N will usually be far less than unity; 
it is an “expected value” in the sense of probability). 
On account of the uniform distribution of track 
angles and their independence of position [(a) and (c) 
of Section 1.4], the average number with track angle 
between (t> and ((> -\- d<t> will be iV'd<^/27r. Fixing our 
attention exclusively on targets of a particular track 
angle 0, it is easy to find how many enter the circle 
per unit time. The relative speed and course are found 
as in Section 1.2 and the circle of radius R is drawn 
about the observer as shown in Figure 8. Since the 
target is moving at velocity w with respect to the ob- 
server, if it is to enter the circle in a unit of time (one 
hour) it must be in the large shaded region of Figure 
8 between the circle and the circle moved through the 
displacement — w, and between their tangents paral- 
lel to w. The area being 2Rw, it is seen that the num- 
ber of targets of track angle between <f) and 0 -t- 
which enter the circle per unit time is (to quantities 


4:RN 

TV 


(u + v)E{a), sin 


(T 


2^/mi 

u -\- V 


Here the second equation results from equation (1), 
the third by introducing a and the new variable of 
integration xf/ = (tt — (f>)/2, and Eia) is the complete 
elliptic integral of the second kind. 

Note that equation (4) is left unchanged if u and v 
are interchanged. This corresponds with the fact that 


/ 

/ 



whenever the target comes within range R of the 
observer, the observer automatically comes within 
this range of the target, etc. 

As an example, if there are 20 vessels distributed 


8 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


at random in an area of 10,000 square miles, so that 
N = 0 . 002 , if they are cruising at 10 knots in various 
directions and if the observer is traveling at 15 knots, 
the number per hour arriving within the range of 
R = 25 miles of the observer is found to be 1.67, 
contrasted with 1.5, which would be the number in 
case the targets remained stationary. That the first 
number is greater than the second is due to the fact 
that the target’s motion tends to bring more of them 
into the range than escape from getting within range 
— a fact which would not have been self-evident 
without calculation. 

The preceding example illustrates the general prin- 
ciple that the contact rate on the random targets 
increases with increase of the motion of the targets. 
To show this, we have but to prove that dNo/du is 
positive. Using the second form for No in (4), but 
with the integral taken over half the interval of in- 
tegration and doubled (a permissible change, in view 
of the symmetry of the integrand), 


dNo 

du 


2RN 

TT 


4 - f Vu^+v^- 

du Jo 


2uv cos (f) dcf), 


2RN n u — V cos <j) 
T Jo W 


m fcoscod^, 

TT Jo 


where co is the angle between the vectors u and v (cf. 
the various cases of Figure 2 , with obvious construc- 
tions). The integrand is always positive when v ^ u 
(cases A and B), so that the required inequality 
dNo /du > 0 is evident. To show that this continues 
to be the case when v > u [case C], decompose the 
interval of integration ( 0 , 7 r) into the two halves 
( 0 , 7 r/ 2 ) and ( 7 r/ 2 , 7 r), and then replace the variable of 
integration in the second half by the supplement of 
the (f) of the first, thus recombining the integrals 

dNo 2RN r'" , 

= ~Jo («os“ + '=os«)d<(.; 

here co' denotes the value which to assumes when 0 is 
replaced by its supplement. Since a simple construc- 
tion based on Figure 2C shows that co' is less than the 
supplement of co, we have cos co' > —cos co; i.e., 
cos CO + cos co' > 0 , and hence dNo/du > 0 , as was 
to be proved. Application: If an enemy is passing 
in our vicinity but along an unknown path, we must 


cut our speed until he passes, if we wish to remain un- 
detected. 

Problem 2. Find the number of targets which enter 
the circle considered above between the bearings ^ 
and /3 + d^, per unit time. Their number is given 
by an expression of the form NQ{^)d^, where iVo(/3) 
is a density related to No by the equation 

No = f No (i8)d/5. (5) 

Jo 

To find Noi^), again we begin by considering only 
those targets of a particular track angle cf). They can 
enter the circle only if the direction of the vector w 
points into the circle, i.e., if the angle y between the 
reversed vector — w and the unit vector n normal to 
the circle and pointing outward is acute (7 is defined 
as measured between 0 and tt). Figure 8 shows that 
in this case the targets in question all come from the 
small heavily shaded region of area Rw cos 7 d^ 
= ( — w-n)M/3 where ( — w-n) denotes the scalar 
product. The number per unit time is obtained by 
multiplying this expression by the density Nd<f)/2Tr. 
Hence, finally, the number No(l3) for targets of all 
track angles is given by 

iVo(i 8 ) =^J' (-'» ■ •a)d<t>, ( 6 ) 


where the integration is over all those values of </> be- 
tween 0 and 27r for which the integrand is positive. 

For the evaluation of ( 6 ), observe that in view of 
the vector equation w = u — v, (Section 1 . 2 ) we have 


— w • n = v • n — u • n 

= V cos ^ — u cos ((f) — /5). 


It remains to insert this value into ( 6 ), then to de- 
termine those values of 0 for which v cos ^ — u cos 
(<f> — > 0 ,and finally to integrate over such values. 

The details of this straightforward computation are 
omitted. As a result, the following expressions are 
obtained. 


When V ^ u: 

No (jS) = cos“^ u 


cos jS 


-\-\/u^ — v^ COS^ jsj. 


(7) 


When V > u: 


u u 

No (jS) = NRv cos jS, when — cos“^ - ^ (3 ^ cos“^ 




UNIFORMLY DISTRIBUTED TARGETS 


9 


No(/3) = 

NRf . / 

V cos~^ 1 — 

V \ 

- cos jS ] cos jS 

TT L \ 

u ) 



+ cos^ jS 

when 

-cqs-‘(-“) 


or when 

cos-‘Qg 

^ = cos-‘(-^); 


iVo(« 

1 = 0, 

when 

/5g-cos-(-H) 

or 

jS ^ cos“ 

■(-T> 


} 


( 8 ) 


This result can be stated in terms of probabilities. 


Suppose that it is known that a target has entered 
the circle; where is it likely to have entered? If 
p{l3)dl3 is the probability that it entered between 
the bearings /5 and jS + d^, the expected number 
No (I3)dl3 which enter per unit time is the number en- 
tering in unit time No times the probability p{fi)d^; 
thus 

NoW 


p(0) = 


No 


iriVoCiS) 


( 9 ) 


ARN {u + v)E (<r) 


in virtue of equation (4). Thus equations (7) and (8) 
give p(/3) at once. 

Figure 9 gives the polar diagram showing the de- 


300 


30 



60 


90 





10 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


pendence of p{^) on for different values of u/v. At 
one extreme, u/v = 0: The targets are at rest and 
the dependence on ^ is as the cosine, and the diagram 
is a circle with the observer at the circumference. At 
the other extreme u/v — co\ the targets move but 
the observer is at rest; in this case the number enter- 
ing at all bearings is the same and we have a circle 
centered at the target. In the intermediate cases, as 
long as target speed u is less than observer speed v, 
there is a certain angular range aft over which no 
contacts are made. As u becomes greater than v, 
however, the number of contacts made aft increases 
rapidly until in the limit as many are made aft as 
ahead. 

There is a sort of inverse to this problem which it 
is useful to consider. Suppose that a contact has 
actually been made at the known bearing jS (and 
range R), what is the distribution of values of the 
track angle (/>? In other words, we have seen the 
target — what is its heading likely to be? This is es- 
sentially a problem in the “probability of causes’’ 
and is solved by Bayes’ formula:® 


n^(<^) = 


^(4>)UW ^ 


where the region of integration must be determined 
as in problem 2, since here again values of for 
which the integrand is negative are excluded. The 
results are, 

when V ^ u, 


(</)) = 2* 


cos ^ cos {<f> — jS) 

V 


cos“^ ( — cos 0) cos + \/ 1 - I ~ cos^ i3 


when V > u, 

= eos^' “J’ 


when 


^ o ^ -1 ^ 

— cos ^ - < < cos ^ 

V~ V 


n^(<^) = 


cos — cos {(j) — i3) 

V 


1 

2 y 

cos~^ ( — cos jS ) cos 13 + 


- ) — cos^ /3 


where Zi(4>)d4> is the “a priori probability” (i.e., as es- 
timated before the contact was obtained) of a track 
angle between and 0 + ^0; /<^(j3) is the “produc- 
tive probability” of the effect observed (i.e., of a 
contact between the bearings ^ and jS + d^), given 
that the target actually has the track angle 0; and, 
finally, n^(<^)d</) is the “a posteriori probability” 
(i.e., as estimated after the contact at bearing ^ has 
been observed) that the target’s track angle lies be- 
tween <f) and <i) + d<l). In other words, n; 3 (</)) is the 
quantity we want. 

As before, 5(<A) = l/27r. To obtain /^(/3)d/3, observe 
that it equals the average number of targets detected 
in unit time between bearings ^ and ^ + d/3, divided 
by the average number detected in unit time at all 
bearings (both averages for targets of given track 
angle </>). This quotient is calculated at once by 
means of the reasoning used before (based on Figure 
8); it has the value 


( — W'n)di8 _ V cos ^ — u cos (</> — /3) 


2w 


2w 




Thus 


n^(</)) = 


(r:w-n)_ 

/( — w • n)d(f)’ 


cSee, for example, Probability and Its Engineering Uses, T. 
C. Fry, D. Van Nostrand Co., New York. 



or when cos“^ - ^ /? ^ cos ^ 

V 

Detection is impossible when 

0 s -cos- (-^) 

or jS ^ cos“^ 

Problem 3. Given the relative position (r,/3) of a 
target at a particular epoch t; find the probability P 
that the target will enter the circle of radius R 
centered on the observer. Find the curves of con- 
stant probability. 

Evidently P depends on (r,|S): P = P(r,^), and 
P = 1 if r ^ i?. When r > R the target will enter 
the circle if, and only if, its vector relative velocity 
w points into the circle (i.e., when w is produced in 
the direction it is pointing). The situation is illus- 
trated in Figures 10 A (v > u) and lOB (v < u), 
which show the angular range of vectors w pointing 




UNIFORMLY DISTRIBUTED TARGETS 


11 


TARGET 




Figure 10. Probability of entering circle. 

into the circle, i.e., the range of angles 6. Corre- 
sponding to this angular range of 6 the angular range 
of <f) is constructed immediately (shaded angle in 


Figure 10). On account of the uniformity of the dis- 
tribution [in particular, of Section 1.4, (a) and (c)], 
the probability that 0 lie in this angular range is the 
magnitude of the range divided by 27r. This is the 
required value of P{r,^). The problem is thus re- 
duced to the geometry of Figure 10, and the formula 
for P{r,ff) is obtained by straightforward trigonome- 
try. There are, however, a number of different cases 
to be considered. For example, in Figure lOB the 
angular range of d is between the two tangents from 
the target to the circle of radius R, while in Figure 
10 A it is between one such tangent and the tangent to 
the velocity diagram circle, corresponding to the re- 
stricted orientations of w when v > u (limiting ap- 
proach angle); moreover, in this case the 6 range 
counts multiply: To one 6 there are two 0’s, one for 
the target moving toward the observer and the other 
for the target moving away and being overtaken. 
There are other cases not shown in Figure 10. 

The expression of P{r,(3) is as follows: 


where is the total radian length of the range 

or ranges of values of 0 (0 ^ 0 < 27 r) which satisfy 
the inequality 

r‘^[u sin {13 — (f>) — v sin = R^lu"^ -{-v^ — 2uv cos 0], 
subject to the condition 

u cos — (f)) ^ V cos /3. 

The second of the above inequalities is needed to in- 
sure that the target enter the circle of radius R after 
the reference epoch t. It is automatically satisfied 
for those values of 0 which satisfy the first inequality 
when V = u. 

The curves of constant probability are symmetrical 
with respect to the course of the observer. In the 
following discussion, only the half-plane to the right 
of the observer’s course is considered. First, consider 
the case when v = u and k = v/u (Figure 11). Out- 
side the circle of radius R and between the tangents 
to this circle which are inclined to the right and the 
left of the course of the observer at the limiting ap- 
proach angle sin“^ u/v, the curve of constant prob- 
ability P is a straight hne tangent to the circle and 
inclined to the observer’s course at the angle sin“^ 
[{u/v) cos TT P]. When v = u, this latter angle reduces 
to {t/2) (1 — 2P). On and below the lower tangent 
line inclined at the angle sin“^ u/v, P = 0. 




12 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


Above the upper tangent line inclined at the angle 
sin~^ ujv^ the equation of the curve of constant prob- 
ability P is 

(jsc^ sin^ — cos^ 

sin^ ^ cos^ ) 

When V = w,this equation reduces to r = P esc {jP/2). 


Second, consider v < u and k = v/u (Figure 12). 
Outside the circle of radius R, the equation of the 
curve of constant probability P is 

sin^ /3 [cos^ (i/' + ttP) — cos^ 

= sin^ + ttP) [u^ cos^ {yp + ttP) — cos^ xp], 

where \p is the positive acute angle cos“^ (PA)* 
Figure 13 shows how the equiprobability curve 
P = 0.25 varies with k = v/u. Figure 14 is for later 
reference. 






UNIFORMLY DISTRIBUTED TARGETS 


13 





14 


POSITION, MOTION, AND RANDOM ENCOUNTERS 



240 


210 


180 


150 


120 


Figure 13 , 25 per cent contact probability curve. 



UNIFORMLY DISTRIBUTED TARGETS 


15 





16 


POSITION, MOTION, AND RANDOM ENCOUNTERS 


16 NONUNIFORM DISTRIBUTIONS OF 
TARGETS 

The uniform distribution of targets considered 
hitherto is by no means the only important case 
which arises in connection with naval operations. 
One example will illustrate this point; it will be used 
later on in the subject of antisubmarine hunts. 

A target has been detected inaccurately. All that 
is known is that it is more likely to be at a certain 
point 0 (the “fix’’) than at any other point, but may 
not be at 0 but only within a short distance of 0 , all 
points at the same distance r from 0 being equally 
likely, and the probability falling rapidly to a neg- 
ligible value as the distance r increases beyond a few 
miles. If f{r)dA denotes the probability that the tar- 
get be in the area dA r miles from 0, the graph of /(r) 
against r will be of the character shown in Figure 15. 


the observation. The target will have moved and the 
distribution will no longer be the same. Assume that 
the speed of the target can be estimated with satis- 
factory accuracy, but not its direction: u is known, 
but not u, i.e., not 0 . It is natural to assume further 
that all directions are equally likely and are inde- 
pendent of the actual position of the target. Thus 
the distribution complies with Section 1.4 (a) and 
(c), for a uniform distribution, but not with Section 
1.4 (b). The problem is to find the new distribution 
f{r,t) after t hours. 

Consider first the case of a target whose vector 
velocity u makes a given angle y with the direction 
from 0 to the contemplated position [7 is measured 
as usual from vector r (from 0 to reference point in 
dA) to vector u]. This target will be in dA if and 
only if it had initially been in a region congruent to 
dA and situated ut miles away in the direction of 



Figure 15. The distribution of targets about a point 
of fix 0 . 


Under such conditions it is almost always possible 
to approximate to the situation with sufficient ac- 
curacy by assuming a circular or elliptical normal 
distribution; we shall assume the former, i.e., that 
/(r) is proportional to where o- is a constant 

(the standard deviation) which increases the more 
the graph of /(r) is spread out, that is, the more in- 
definite the knowledge of the target’s position. The 
constant of proportionality is found by the fact that, 
since the target is surely somewhere, the integral 

extended over the whole 

surface has the value unity. Thus one obtains 


Jj f{r)dA ( = JJ* f{r)rdrdyj)j 


fir) = 


1 

2Tr(j^ 




Now suppose that t hours elapse after the time of 


dA INITIALLY 



the reversed vector — u, as shown in Figure 16. The 
probability of this event is 

^ p - A _ L_ ^ -(r* + uH^ - 2rut cos y) 

27rcr2 27r(T2 


Now the probability that u make an angle with r 
between 7 and 7 ^7 is dyl2T. The probability of 

both these events is the product of these two prob- 
abilities, and to obtain the required total probability 
f{r,t)dA this product is added (integrated) over all 
possible initial positions of dA, i.e., over all values of 
7 from 0 to 27r: 

f(r,t) = Ar%-(r>+»>l"- 2 ™(coST)/ 2 .->rfT, 

Ztto- ZTjf. 


2Tra^ 


g-(r2 +m2<2)/2o.2 


2do 


g2ru< cos y/2o^^y^ 






NONUNIFORM DISTRIBUTIONS OF TARGETS 


17 


Now we have 


1 f>2ir 1 p2Tr 

I ^rut cos y/<r^^y — I rut cos y/a^^y 

2Tjo 27rJ„ 


1 p2Tr 

_ I Qi{irut) cos y/<r'^(Jly 

2^ Jo 


- / . rut\ T ( rut\ 

- ■’{' If) - 'if} 


where i denotes the ordinary Bessel 

function of zeroth order, lo its value for pure imag- 
inary values of the argument. Thus the equation 

^ + ( 10 ) 

The graph of f(r,t) for different values of t is shown 
in Figure 17. It is seen how the probabihty spreads 
outward with time, so that the target is most likely 
to be in an expanding ring about 0. 



Figure 17. The distribution of moving targets about a point of fix 0. 


Chapter 2 

TARGET DETECTION 



ERALITIES CONCERNING 
DETECTION 


T he first chapter has dealt with the positions, 
motions, and contacts of observer and target, but 
has left entirely out of consideration the act or process 
whereby the observer gains knowledge of the presence 
and position of the target. Contacts have been con- 
sidered as purely geometrical events and their prob- 
abilities have been simply the probabilities that the 
target reaches a specified position in relation to the 
observer. The present chapter will be concerned, on 
the other hand, with the act of detection, that event 
constituted by the observer’s becoming aware of the 
presence and possibly of the position and even in 
some cases the motion of the target by visual sight- 
ing, by radar detection, by hydrophonic listening, by 
echo ranging, or by any other means whatsoever. 
There are certain general ideas and methods which 
apply to all cases of detection, and their study is the 
object of this chapter. But before quantitative re- 
sults of immediate practical applicability can be ob- 
tained, a detailed study must be made of the special 
instrumentality of detection; this is done for the 
visual case in Chapter 4, for radar in Chapter 5, 
and for sonar in Chapter 6. 

Two basic facts underlie every type of detection: 
(i) There is a certain set of physical requirements 
which have to he met if detection is to he possible, and 
which if met will in fact make detection possible, though 
not necessarily inevitahle. Thus targets must obviously 
not be too far away: their view from the observer 
must not be completely obstructed; to be seen there 
must be some illumination; the radar will not reveal 
them if the atmospheric conditions or background 
echoes are too bad ; sonar detection requires that the 
sound path be not completely bent away from the 
observer by water refraction; etc. 

(ii) Even when the physical conditions make detection 
possible, it will by no means inevitably occur: Detection 
is an event which under definite conditions has a definite 
probability, the numerical value of which may be zero 
or unity or anything in between. Thus when the target 
just barely fulfills the physical conditions for possible 
detection, the probability of detection will be close 
to zero (at least when the time for observation is 


very limited). As the conditions improve the chance 
of detection increases, and it may become close to 
or equal to unity : detection becomes practically cer- 
tain. Experience in everyday life shows that we may 
be looking for an object in plain sight and yet some- 
times fail to find it. Cases are known where obser- 
vational aircraft flying on clear sunny days on ob- 
servational missions have passed close over large 
ships and yet failed to detect them. And a host of 
operational statistics give further confirmation of 
this point. It must be constantly realized that every 
instrumentality of detection is based in last analysis 
on a human being, and its success is accordingly in- 
fluenced by his attention, alertness, and fatigue, and 
the whole chain of events which occur between the 
impact of the message on his sense organs and his 
mental response thereto. Furthermore, even under 
physical conditions which are as fixed and constant 
as it is practicable to make them, innumerable rapid 
fluctuations in them are still apt to occur (a radar 
target changes its aspect from moment to moment 
with the continual rocking of the ship, sonar ranges 
experience short-term oscillations about their mean, 
etc.); and as ^ result, a target which may not be de- 
tected at one instant may be detected if sought a 
moment later. 

In view of (i), one part of the study of detection 
requires the physical conditions for detection to be 
explored; in view of (ii), the other part requires the 
probabilities of detection, when the former conditions 
are given, to be obtained. 

In Section 2.9 the effect of statistically combining 
observations made under operational conditions in 
which the physical situation is not constant is con- 
sidered. 


22 INSTANTANEOUS PROBABILITIES OF 
, DETECTION 

Suppose that the physical conditions (distances, 
etc.) remain fixed and that the observer is looking 
for the target (by “looking” shall be meant trying 
to detect with the means considered, visual, radar, 
sonar, etc). There are two possibilities: First, the 
observer may be making a succession of brief 


INSTANTANEOUS PROBABILITIES OF DETECTION 


19 


“glimpses”; a typical case of this is in the echo- 
ranging procedure in which each sweep or scan affords 
one opportunity for detection (glimpse), successive 
ones occurring two or three minutes apart. Second, 
the observer may be looking continuously; a typical 
case is the observer fixing his eyes steadily on the 
position where he is trying to detect the target. The 
case of radar is intermediate; on account of the scan- 
ning it would belong to the first case, but if the 
scanning is very fast, and especially when there is 
persistency of the image on the scope, it may be 
treated as in the second. Likewise, visual detection 
by a slow scan through a large angle belongs to the 
first rather than the second case. Very often the 
decision to regard a method of detection in the first 
or in the second way depends simply on which affords 
the closest or most convenient approximation. This 
will be made clear on the basis of examples in Chap- 
ters 4, 5, and 6 . 

In the case of separated glimpses, the important 
quantity is the instantaneous 'probability g of detection 
by one glimpse. When n glimpses are made under 
unchanging conditions the probability Pn of detec- 
tion is given by the formula 

= 1 - (1 - g)"- ( 1 ) 

This is because 1 — is the probability of failing 
to detect with n glimpses, and for this to occur the 
target must fail to be detected at every single one of 
the n glimpses; each such failure having the prob- 
ability I — g and the n failures being independent 
events, we conclude that 1 — Pn = (1 — g)^', hence 
(1). When g = 0, obviously = 0, but if gf > 0 and 
even if g is very small, can be made as close to 1 as 
we please by increasing n sufficiently; in other words, 
once the physical conditions give some chance, how- 
ever small, of detecting on one glimpse, enough 
glimpses under the same conditions will lead with 
practical certainty to eventual detection. 

To find the mean or expected number n of glimpses 
for detection we must first find the probability P„ 
that detection shall occur precisely at the nth glimpse 
(and not before). This is the product of the prob- 
ability that it shall not occur during the first n — 1 
glimpses, (1 — gf)”~^ times the probability that a de- 
tection shall occur on a single glimpse (the nth), g; 
it is accordingly Pn = (1 — The required 

mean number n is, according to the theory of prob- 
ability, IPi + 2P2 + 3P3 + • • • , and thus 


00 

n = - gy-'g 

n = 1 

= S+2(l-6i)sf + 3(l-3)^9+ ••• 

= - g^l + (1 - 3) + (1 - 9)2 + . . . j (2) 

d 1 
^dgl-{l-g) 



1 

g 


Turning to the case of continuous looking, the im- 
portant quantity is the probability ydt of detecting in 
a short time interval of length dt. The quantity 7 is 
called the instantaneous probability density (of de- 
tection). When the looking is done continuously dur- 
ing a time t under unchanging conditions, the prob- 
ability p{t) of detection is given by 

p(0 = 1 - (3) 

To prove this, consider q{t) = 1 — p{t), the prob- 
ability of failure of detection during the time t. For 
detection to fail during the time t + dt [probability 
= q{t dt)], detection must fail both during t [prob- 
ability = q{t)] and during dt (probability = 1 — ydt), 
and multiplying these probabilities of independent 
events we obtain 

q{t + dt) = q{t)(l - ydt) 
which is equivalent to the differential equation 

dt 

The solution of this equation on the assumption that 
g( 0 ) = 1 (no detection when no time is given to look- 
ing) is q{t) = e~'^^\ whence (3). Again it is true that 
if there is the least chance of detection in time dt 
(i.e., if 7 > 0 ) the chance of detection increases to 
virtual certainty as the looking time t becomes suffi- 
ciently large. It may be observed that the quantity 
yt in the exponent of (3) represents the mean or 
expected number of targets detected by an observer 
passing through a swarm of unit density of targets 
uniformly distributed over the ocean. 

To find the mean or expected time t at which de- 
tection occurs, observe that the probability P{t)dt of 




20 


TARGET DETECTION 


detection between t and t + dt (when looking has 
been continuing from the initial time 0) is the product 
of probability of no detection before t times prob-. 
abihty of a detection during dt, i.e., P(t)dt = ydt. 
i is found by integration 



Figure 1 shows the graphs of the probability p(t) 
of detection during the time t and P{t) of detection 
at the time t and gives the construction of t as the 
abscissa of the intercept with the horizontal line of 
unit ordinate of the tangent to p{t) at the origin. 

Since equation (3) reduces to equation (1) when 7 
is taken as 7 = — log (1 — g) and t = n (glimpses one 
unit of time apart), Figure 1 serves to show the quan- 
titative behavior of Pn and : the difference is that 
only discrete points (^ = 1, 2, 3, • • •) on the curve 



Figure 1. Probabilities of detection under fixed 
conditions. 


are used, and n is no longer given by the tangent in- 
tercept but rather by a secant intercept.^ 

When, as usually occurs in actual search, the dis- 
tances and hence the probability quantities g or y 
change as time goes on, (1) must be replaced by 

p„ = 1 - n^d - gO = 1 - (1 - sOd - g,) 

d - 6f3) • • • (5) 

which takes into account the fact that g will change 
from glimpse to glimpse: g; is the probability of de- 
tection for the ith glimpse. And (3) must be replaced 

by 

p(t) = 1 - (6) 

where in yt the possible change in the probability 
density of detection as time goes on is put into evi- 
dence by the subscript. The reasoning leading to 
these equations is precisely similar to that in the 
earher case. But the probabilities Pn, p{t) do not 
necessarily approach unity as n or ^ increase, thus 

^Throughout this book, log is used to denote “natural log- 
arithm,” and logio to denote “common logarithm.” 


when /“ ytdt is finite, the chance of detection p{t) 
never exceeds 1 — e~'^" < 1. Here again the 
quantity /“ ytdt in the exponent represents the ex- 
pected number of targets detected as the observer 
passes through a swarm of targets distributed uni- 
formly at unit density over the ocean. 

The instantaneous probabilities quantities g and y 
depend, as we have said, on the sum total of physical 
conditions. For example, in visual detection y de- 
pends on the range r from target to observer, on the 
meteorological state (illumination and haze), on the 
size and brightness of target against the background, 
on the observer's facilities, altitude, etc. And corre- 
sponding lists can be made out for radar and sonar 
detection. Throughout the remainder of the present 
chapter, only the dependence on range will be ex- 
plicitly considered, i.e., we shall write 

g = sW. T = y(r). (7) 

It will be legitimate to apply the results either when 
all the other conditions remain practically unchanged 
during the operation considered, or when the other 
conditions have been shown not to influence the re- 
sults to the degree of approximation that is accepted. 

Since the instantaneous probability quantities 
tend to decrease to zero as the range r increases and 
to be large when the range is small, their graph 
against r will be of the character shown in Figure 2. 
Case A is when the instantaneous probability density 
reaches a finite maximum at zero range (probability 
of detecting target when flying over target is less 
than unity). In case B this maximum is infinite (prob- 
ability of detection when flying over target is unity) . 
In case C the effect of sea return on radar diminishes 
the probability of detection when over target. In 
case D, the instantaneous probability is infinite when 
r < R: detection is sure to occur as soon as the tar- 
get gets within this critical range R. 

The last case, while not altogether realistic, is often 
not very far from the truth. A very useful rough ap- 
proximation is to assume further that the instantane- 
ous probability is zero for r > R. Then detection is 
sure and immediate within the range R and is im- 
possible beyond R. This assumption shall be called 
the definite range law of detection. 

An important example showing the evaluation of 
the function y{r) is in the case when the following as- 
sumptions are made. 

1. The observer is at height h above the ocean, 
on which the target is cruising. 


muEimmniu 


INSTANTANEOUS PROBABILITIES OF DETECTION 


21 





length a toward the observer and width h perpen- 
dicular to the direction of observation (perpendicular 
to the page in Figure 3 A). The infinitesimal solid angle 
is the product of the angle a subtended by a, and 
the angle ^ subtended by h. The radian measure of 
a is c/s. By similar triangles, c/a = h/s and hence 


O OBSERVER 



0 



Figure 3. Solid angle subtended by wake. 



Figure 2. Instantaneous probability at various distances. 


a = ah/s^. And the radian measure of ^ is obviously 
h/s. Hence solid angle = ajS = ahh/s^ = area of 
rectangle times h/s^. The actual area A of the tar- 
get’s wake is not rectangular, but can be regarded 
as made up of a large number of rectangles like the 
above, the solid angle being the sum of the corre- 
sponding solid angles. Hence, when the dimensions 
of A are small in comparison with h,r, and s, we have 
the formula 


Solid angle = — 

5 ® 


Ah 

(/i2 + r2)^’ 


( 8 ) 


2. The observer detects the target by seeing its 
wake. 

3. The instantaneous probability of detection y 
is proportional to the solid angle subtended at the 
point of observation by the wake. 

The calculation of the solid angle is shown in 
Figure 3 for an area of ocean which is a rectangle of 


Since y is assumed to be proportional to the solid 
angle, we obtain 

kh kh /-X 

y = — = -3> (9) 

where the constant k depends on all the factors which 
we are regarding as fixed and not introducing ex- 




22 


TARGET DETECTION 


plicitly, such as contrast of wake against ocean, ob- 
server’s ability (number of lookouts and their facil- 
ities), meteorological conditions, etc.; and of course k 
contains A as a factor. Dimensionally, k = 

In the majority of cases r is much larger than h, and 
(9) can be replaced by the satisfactory approxima- 
tion 

. = ( 10 ) 


The latter need not be moving in fixed course and 
speed over the ocean, although that is very often 
the case. The coordinates used have been described 
in Section 1.2 (see Figure 4 and equation (2) of 
Chapter 1. 

The target is at {^,ri) at the time t, so that the 
equations of the target’s relative motion are 

^ = m, V = vit), ( 11 ) 


Formulas (9) and (10) lead to cases A and B 
respectively of Figure 2; the property of detection 
which they express shall be called the inverse cube 
law of sighting. When the subject of vision is studied 
in Chapter 4 it will be found that many changes in 
this law have to be made to obtain a high degree of 
approximation under the various conditions of prac- 
tice. Nevertheless the inverse cube law gives a re- 
markably useful approximation. Its use in the 
present chapter is chiefly as an illustration of the 
general principles. 


23 DEPENDENCE OF DETECTION ON 
TRACK 

When the observer and target are moving over the 
ocean in their respective paths, which may be straight 
or curved and at constant or changing speeds, the 
continuous change in their relative positions con- 
stantly changes the instantaneous probability of de- 



Figure 4. Target’s relative track. 


where initially {t = t') = ^(^Oj ^70 = 'nil'), and 

finally {t = t") = 77 (^"). The target de- 

scribes the relative track C. Accordingly, (7) becomes 
[writing ^^(Q for { ^(Q ) 2 ] : 


g = + V(0) = gt 

7 = 7(VfH0 + = 7(- 


( 12 ) 


Hence according to equations (5) and (6) the prob- 
abilities Pc of detection are given by either of the 
following 

Pc = 1 -.n [1 - g , (13) 

Pc = 1 - exp + »)^(<))d<J. (14) 

In (13) ti is the time (epoch) of the ?ith “glimpse” or 
scan, and n the number of glimpses between t' and f: 

t' ^ti<t2< • • • <tn^ I". 

In (14), the integral is actually a line integral along 
C; if w is the relative speed (not necessarily con- 
stant), we may write [with s = arc length of C from 
(^0,^70) ]• 

Pc = 1 - (15) 


Formulas (13) and (14) may be united into 


Pc = I - e 


-F[C] 


(16) 


where for the case of separate glimpses 

n 

F[C] = -Ilog[l - g{yi^{l,) + ,H1,.))], (17) 

i = \ 


tection; we have to deal with the functions gt and and in the case of continuous looking 
yt and calculate probabilities of detection by means 
of formulas (5) and (6). It is convenient to draw the 
target’s track C (Figure 4) relative to the observer. 



DEPENDENCE OF DETECTION ON TRACK 


23 


This quantity F[C] shall be called the sighting 'po- 
tential. It has the important property of additivity: 
If Cl and C 2 are two tracks and C = Ci + C 2 is their 
combination or sum, and if pc = Pci + Ci Is the prob- 
ability of sighting on at least one track, + <72 is 
still obtained by formula (16) and 

F[C, + C 2 ] = F[Ci] + FIC 2 \. (19) 

This is an immediate consequence of the usual equa- 
tion for combining probabilities of events which may 
not be mutually exclusive: 

Pc = I - (I - Pc^) (1 - Vc.) = Pc. + PC2 - PCxPC2- 

The additivity applies, of course, to the sum of any 
number of paths. One application is to the calcula- 
tion of Pc when C is complicated, but made up out of 
a sum of simple pieces such as straight lines. Another 
application is in the case of two or more inter-com- 
municating observers: Ci can be the path of the target 
relative to the first and C 2 that relative to the second, 
etc. 

A most important case, and one which will chiefly 
concern us in this book, is when both observer and 
target are moving at constant speed and course. The 
results of Chapter 1 become applicable. Track C is 
a straight line, and the speed it; is a constant (as long 
as C is not turned). It is convenient to make the 
calculations with the aid of the coordinates {x,y) of 
Chapter ] (Figure 5), where x is the lateral range. 
The equations of motion which take the place of (11) 
are x = constant, y = wt, where t is measured from 
the epoch of closest approach, and where, furthermore, 
the positive direction of the y axis is that of the tar- 
get’s relative motion; this convention is used through- 
out this chapter. The potential F[C] is given by the 
appropriate one of the formulas 

P\c] = — ^ log 1^1 — g(;Vx‘‘ + 

( 20 ) 

= - 2!) “ 6t(\/ + ?/,■*)] ; 

1=1 



F[C] = I 7 (Vx^ + wH^)di, 

1 ry’ (21) 

^ w I '^{Vx^ + y^)dy, 

J y' 

where yi is the distance of target at the fth glimpse 
to its closest position, and {x',y') and {x",y") are the 



a )-< 0 , a )">0 

A 



Figure 5. Detection at fixed speed and course. 




X I 


24 


TARGET DETECTION 


extremities oi C: x' = x'' = x = constant, y' = wt\ 
y" = wt\ 

In the case of the inverse cube law (9), 


FlC] 


W Jy' 


dy 


m 


h^ + 


{h-^ + + 2/2)^ 

-(-=JL= 

^\Vh'^ + + {y'y 


( 22 ) 


+ {y') 




And for (10), 

kh dy 

y 


v\n - ^ - -(y^ - 

^'~wj^.{x^ + y^y~x\r'' r'J 


= ^(sin a' + sin 0 ,"), 


where in each case 


kh 

m = — , 
w 


(23) 


(24) 


and where r' and r" are the ranges of the extremities 
of C, and co' and co" the angles they subtend with the 
normal to C. 


2^ THE LATERAL RANGE DISTRIBUTION 

When the observer and target are on their straight 
courses at constant speeds for a long time before and 
after their closest approach, the probability p(x) of 
detection is a function of the lateral range x. The 
graph of p{x) against x is called the lateral range curve 
and expresses the distribution in lateral range. In con- 
sequence of (16) p{x) is given by 

p(x) = 1 - (25) 

where F{x) is the value of F[CJ, C* being an in- 
finite straight line at the perpendicular distance x 
from the observer. The value of F{x) is found by 
applying equation (20), summing over all integral 
values of f,or equation (21) with y' = —co and ?/" = «» , 
in the glimpse or the continuous looking cases, re- 
spectively. 

With continuous looking (21) applies. For the 
definite range law, p{x) = 1 or 0 according as 
— R<x<R or not, and the lateral range curve 
is Figure 6A. For the inverse cube law, 

p(x) = 1 - e”^”''‘‘’+“">orp(x) = 1 - (26) 


according to whether (22) or (23) is used; the curve 
is shown in Figure 6B in the former case. 

With intermittent glimpses taking place T units of 
time apart, equation (20) applies. For the definite 
range law, p(x) =0 when x > R or x < —R, and 
p{x) = 1 when the length 2\/r^ — relative track 
during which the target is within range R of the 
observer is greater than wT, i.e., when 


but 


-V S X S - M)2rV4 ; 

2V - x^ 


p(x) = 


wT 


in intermediate cases, this being the probability that 
the target be glimpsed while within range R. The 
lateral range curve is shown in Figure 6C. 

Other typical lateral range curves are those of 
Figure 6D and E. The dip at a; = 0 in Figure 6E 
shows the effect of sea return (radar) or pinging over 
the target (sonar). 

The area W under the lateral range curve is called 
the effective search {or sweep) width : 

W = f^ p{x)dx. (27) 


It has the following interpretation. If the observer 
moves through a swarm of targets uniformly dis- 
tributed over the surface of the ocean {N per unit 
area on the average) and either all at rest or all 
moving with the same vector velocity u, the average 
number Nq detected per unit time is 

Ao = NwW. (28) 

For suppose that t is such a long period of time that 
the length of time during which a target is within 
range of possible detection is small in comparison 
with t. Then the number of targets passing during 
the period t through detection range (i.e., exposing 
themselves to detection) and having the lateral range 
between x and x dx is Nwtdx (since such targets 
are in an area of wtdx square miles). On the average 
p{x)Nwtdx of these will be detected. Hence the aver- 
age total number detected is 

^+00 

I p{x) Nwtdx. 


Dividing this by t and applying equation (27), equa- 
tion (28) is obtained. Since for continuous looking 
with a definite range law, W = 2R, we may describe 
W as follows: 

The effective search width is twice the range of a 




THE LATERAL RANGE DISTRIBUTION 


25 


definite range law of detection which is equivalent to 
the given law of detection in the sense that each of the 
two laws detects the same number of uniformly dis- 
tributed targets of identical velocity. 

The product wW is called the effective search {or 
sweep) rate. 

When the distribution of targets is uniform in the 
sense of Section 1.3, i.e., when their speed is given 


so that the search width is proportional to the square 
root of the altitude and inversely proportional to the 
square root of the target’s relative speed. Further- 
more, if there are n aircraft flying the same path 
without mutual interference (or if there are n ob- 
servers having the same facilities operating inde- 
pendently of one another in the same aircraft), W is 
replaced by W -y/n^. 


A 

1 

P(x) 

B 1 

X — 

P(X) 

-R 

R 






but their course is not, w has to be replaced by its 
average w (taken as uniformly distributed in track 
angle </>) i.e., we must write w = so that 

(28), No = NwW, may hold {W = 2R). [See Chap- 
ter 1, equations (1) and (4)]. 

In the case of the simplifled inverse cube law 
equations (26) and (27) give, by carrying out the in- 
tegration (see below) : 

W = = 2./—, 

\ w 


This results from the additivity of the potentials 
Section 2.3, which has the effect that k is replaced 
by nk in equations (22) and (23), and thus that m is 
replaced by nm [see (24)]. Thus the statement that W 
is replaced by IF ^/n is a consequence of equation (29). 

The integration leading to (29) is performed by 
introducing equation (26) into (27) and changing 
to the new variable of integration: 

\/2m . 


(29) 


z 


X 


26 


TARGET DETECTION 


and then integrating by parts. Use is made of the 
well-known equation 



By its definition, p{x) is the probability {not prob- 
ability density) that a target, known to have the 
lateral range x, be detected. On the other hand, 
'p{x)dx/W is the probabihty that a target, known 
to have been detected, have a lateral range between 
X and X -\- dx (in this case p{x)/W is sl probability 
density). This fact (actually a consequence of Bayes' 
theorem in probability) is easily seen, as follows: 
The detected target may be thought of as chosen 
at random from the set of all detected targets; the 
chance that its lateral range be between x and x dx 
is equal to the proportion of targets in this set which 
have such a lateral range; from the previous calcula- 
tions, this proportion is seen to be 

Nwp{x)dx _ p{x)dx 
NwW 


Np{x,y) dxdy, where p{x,y), which may be described 
as the rate of first contacts at the point {x,y) per unit 
area and per unit density of targets, is obtained by the 
argument which follows. 

The number of targets entering dxdy in unit time 
is Nwdx. A given target’s probabihty of being de- 
tected therein is the product of the probability that 
it fail to be detected before entering this region times 
the probability that, when not previously detected, 
it be detected while crossing dxdy, i.e., during the 
time dt = dy/w. The former probability is in 

virtue of equation (16), where F{x,y) is given in the 
case of glimpses by equation (20), with i summed 
over all values for which yi < y (with sufficient ac- 
curacy we may write yi = y — iwT and sum for 
^ = 1, 2, • • • oo); and in the case of continuous 
looking, by equation (21) with y' = — co and y" = y. 
The latter probability is given by g{r)dy/wT (inter- 
mittent glimpses, one every T units of time, dy/wT 
being the probability that a glimpse occur while 
target is in dxdy), or by y{r)dy/w (continuous look- 
ing). Thus the probability is 


25 THE DISTRIBUTION IN TRUE RANGE 

Again we suppose that the observer makes con- 
stant speed and course and that the targets do like- 
wise and are distributed uniformly over the surface 
of the ocean with the density N (average number 
per unit area). Relative to the observer, the targets 
all move parallel to the y axis in the direction of 



increasing y. How many targets are detected on the aver- 
age per unit time in the small region of area dxdy of 
Figure 7f The number will be proportional to N 
and to dxdy, and may accordingly be represented by 


e-PMg{r)dy 


e-PMy{r)dy^ 

w 


according to whether glimpsing or continuous look- 
ing is used. To obtain the mean number of detections 
per unit time in dxdy, these expressions are multi- 
plied by the number of targets exposed to such de- 
tection, Nwdx. Hence the answer to the question in 
italics above is supplied by the following expressions 
for p{x,y ) : 


P {x,y) = 


T 


(30) 


F{x,y) = - log[l - g(y + {y - iwry)"] 

i = l 


for intermittent glimpsing, and 
— e~^^^'y^y{r) 

F{x,y) = 7 (Vx^ + y^) dy 


(31) 


for continuous looking. 

It is seen by carrying out the differentiation that 
in the case of equation (31), 

p{x,y) = (32) 




THE DISTRIBUTION IN TRUE RANGE 


27 


In the case of (30), the corresponding formula is 

p(x,y) = wAy[l - (33) 

where the operation Aj, applied to a function denotes 
the result of the follomng process: first, replace y 
in the function hy y Ay, {Ay = wT); second, sub- 
tract the original value of the function from the 
new; third, divide by Ay. 

If A is a plane region moving with the observer 
over the ocean, the average number of targets 
detected per unit time within A is (by addition of 
averages) 

Qa = ^ ff p{x,y)dxdy. (34) 

A 

In particular, when A embraces the whole plane, 
equations (32) and (33) lead from equation (34) to 
the previously obtained expression Nq = NwW of (28) 
by straightforward calculation. When A = is a 
circle of radius R centered on the observer, (34) ex- 
pressed in polar coordinates (r,^ (^ = angle from 
positive y axis to vector r drawn from observer to 
. target) becomes: 

Q{R) = N J'j' f>{x,y)dxdy 

= iV I I rp(r sin f, r cos 0 
Jo Jo 

Now the number of targets detected per unit time 
at a distance (true range) from the observer between 
r and r + dr is of the form Np{r)dr (being propor- 
tional to both N and dr), and since its integral from 
0 to R must, for every value of R, be equal to Q{R), 
it follows (by equating the two integral expressions 
for Q{R) and differentiating through with respect to 
R, etc.) that 

p(r) = I rp(r sin f, r cos f) d^. (35) 

Jo 

p{r)dr may be described as the rate of detection in 
the range interval (r, r + dr) at unit target density. 

If, now, a target is known to be detected but at 
unknown range, the probability that the range of de- 
tection has been between r and r + dr is p{r)dr/wW. 
For this target may be thought of as chosen at 
random from the set of all the NwW detected tar- 
gets, of which there are Np{r)dr detected at range be- 
tween r and r dr. Hence the probability that the 
target be detected at such a range is the quotient 
Np{r)dr p{r)dr 
NwW ^ InW^' 


The function p(r) (or the equivalent functions 
Np{r) or p{r)/wW) expresses the distribution in {true) 
range, and the graphs of these functions against r 
are called range curves. They fall considerably for 
small values of r, since relatively few targets come 
close to the observer by chance, and of these a still 
smaller number are apt to survive undetected up to 
a close proximity of the observer. Figure 8 shows a 
typical range curve (actually, for the inverse cube 
law); as the situation approaches the definite range 
law, the curve humps up indefinitely about the value 
r = R oi the definite range, and falls to the axis of 
abscissas elsewhere. 

The mean value of the range of detection is given by 


r 




sin r cos ^) df 
(36) 


in all cases. 

In the case of the simplified inverse cube law, we 
have equation (23), in which we set co' = 7r/2 — f 
and cu" = 7r/2; we obtain 


m 


^ [^v\ = 3 (1 + cos 


m „ f 
2r2^®^ 2 



Thus 


p{x,y) = w exp 


( 


2r2 


csc^ 


2/ r^ 


Hence’^ 

pW = ^ ^ 2) 

bThis depends on the evaluation of the integral 

This is done by the following device: Differentiating with 
respect to X, 

4>'{X) = — csc2 d csc^d dd 

= e -X r® e -X cot2 e d cot 6 

Jco 



as is seen on changing to the variable of integration x = 
cot d, and the use of the formula Jg" dx = V^/2. In- 
tegrating 0' (X) with respect to X, observing that <^(0) = ir/2, 
we obtain 

</>(X) = i - i/x ^ f [1 - erf /x ] , (38) 

as it appears on changing to the variable of integration /x = i/^. 




28 


TARGET DETECTION 


where “erf X’' (the “error function” or “probability 
integral”) is defined as 

2 

erfX=^| e-^’dx. 

Jo 

By expressing m in terms of the search width W by 
means of equation (29), equation (37) is reduced to 

This is the function actually graphed in Figure 8. 



Figure 8. True range curve. 


For values of r not over about 15 miles, it is in reason- 
able rough agreement with operational data (visual) ; 
but farther out it has too high an ordinate. 


26 RANDOM SEARCH 

In the last two sections both observer and target 
were on straight courses at constant speeds; this 
represents the extreme of simplicity of paths. At 
the other extreme is the case where both are moving 
in complicated paths over the ocean and at speeds 
which may vary in the course of time, a case which 
is called that of random search. If the position of the 
target is in the area of interest A (which may be 
many hundred square miles) in which the observer 
is moving, and if the observer is without pre- 
knowledge indicating that the target is more likely 
to be in one part of A than in another, a good ap- 
proximation to the probability p that the observer 
make a contact is given on the basis of the following 
three assumptions: 

1. The target’s position is uniformly distributed 
in A. 

2. The observer’s path is random in A in the sense 
that it can be thought of as having its different (not 
too near) portions placed independently of one an- 
other in A. 


3. On any portion of the path which is small rela- 
tively to the total length of path but decidedly larger 
than the range of possible detection, the observer 
always detects the target within the lateral range 
IF/2 on either side of the path and never beyond. 

These assumptions lead to the formula of random 
search 

p = 1 — (40) 

where A = area in square miles, IF = effective search 
width in miles, L = total length of observer’s path 
in A in miles. 

To prove this, suppose that the observer’s path L 
is divided into n equal portions of length L/n. If n 
is large enough so that most of the pieces are ran- 
domly related to any particular one, the chance of 
failing to detect during the whole path L is the 
product of the chances that detection fail during 
motion along each piece. If, further, L/n is such that 
most of the pieces of this length are practically 
straight and considerably longer than the range of 
detection, then in virtue of (3) the latter chance of 
detection is the probability that the target be in the 
area swept (whose value is WL/n square miles), and 
this probability is WL/nA [assumption (1)]. Hence 
the chance that along all of L there be no detection 
is (1 — WL/nA)^, and hence 



= 1 — for large n. 

This reasoning assumes, of course, that a large n 
having these properties exists. This is essentially as- 
sumption (2). 

If the exponential in equation (40) is replaced by 
the first two terms in its power series expansion, the 
equation is replaced by p = IFL/A; this corresponds 
to the probability in the case that L consists of a 



single straight line, or a path so little bent that there 
is practically no overlapping of swept regions: The 
total area swept is IFL and the chance of the target 
being in it is IFL/ A. The departure from this simple 




PARALLEL SWEEPS 


29 


value represents the effect of random overlapping of 
swept areas. 

Figure 9 shows the way in which the probability 
increases with the length of observer’s path L. For 
smaller values of L it is closely approximated by its 
tangent p = WL/A. For much larger values, it ap- 
proaches unity, exhibiting a “saturation” or “dimin- 
ishing returns” effect. 

27 PARALLEL SWEEPS 

Search by parallel sweeps is a method frequently 
employed, and many apparently more complicated 
schemes turn out to be equivalent to parallel sweeps, 
either exactly or with sufficient approximation for 
practical purposes. A target is at rest on the ocean in 
an unknown position, all equal areas having the same 
chance of containing it; it is decided to search along 
a large (“infinite”) number of parallel lines on the 
ocean, their common distance apart, or sweep spac- 
ing, being S miles; what is the probability P{S) of 
detection? Or again, the target’s speed and direction 
are known, but the position is uniformly distributed 
as above; it is possible to search in equally spaced 
parallel paths relative to the target, i.e., in the plane 
moving with the target’s motion and in which it ap- 
pears to be a fixed point, as in the first case. It is 
immaterial whether all the parallel paths are trav- 
ersed by the same observer or by different observers 
having similar observing characteristics. 

In Figure 10 the parallel paths are shown referred 
to a system of rectangular coordinates; the axis of 





y 

»(x,y) 




-2S 

-s 

0 

* s 

2S 

3S 

X 


Figure 10. Parallel sweeps. 


ordinates is along one of the paths and the target’s 
(unknown) position is at {x,y), and 0 ^ a; < *S. It is 
observed that this inequality, expressing the fact 
that the target is in the strip immediately to the 
right of the axis of ordinates, is a consequence of the 
method of choice of the axes, and implies no restric- 
tion in the position of the target. 

The first step in calculating P{S) is to write down 


the lateral ranges of the target from the various ob- 
server paths. For paths at or to the left of the axis 
of ordinates, the lateral ranges are 

X, X + S, X + 2aS, x + 3*S, • • • 

For those to the right, 

S — X, 2S — X, 3S — X, • • • 

All these cases may be combined into the absolute 
value formula: 

lateral range = | a: — 1 (41) 

where 

0 ^ a: < and rt = 0, + 1, + 2, + 3, • • • . 

Equation (25) is now applied to find the prob- 
ability Pn = V (^th lateral range) of detection by the 
nth sweep when the target’s position is given as {x,y) 

= 1 - , 

where the potential F is given by the appropriate 
formula. The probability of no detection by the nth 
sweep is 1 — Pnl that of no detection by any sweep 
is the (infinite) product n(l — pn) for all values of 
n ( < 0, = 0, > 0) ; and the probability that at least 
one sweep detect a target given at {x,y) is 

P(x,S) = 1 - n 

n=— 00 

or, finally, 

P{x,S) = 1 - 

+« (42) 

4>(a:,S) = 2] F(|x - nS|), 0 g x < S. 

n = —00 

This is essentially a repetition of the argument prov- 
ing the additivity of the potentials Section 2.3. 

It remains to find the probability of detection P{S) 
when the target’s position (the value of x) is not 
given, but has a uniform distribution between 0 and 
S. An easy probabiUty argument shows that P(S) is 
the average of P{x, S) over all values of x in this 
interval : 

P{S) = dx 

( 43 ) 

<i>(x,S) = H Fi\x - nS\). 

n=— 00 

This gives the general solution of the problem. 

The effective visibility E is defined as half that sweep 




30 


TARGET DETECTION 


spacing for which the probability of detection by 
parallel sweeps is one half. In other words, E is 
determined as the solution of the equation 

P(2E) = i- 


Three cases are of particular interest. The first is 
that of continuous looking on the assumption of a 
definite range law. Detection will surely occur if, 
and only if, the target happens to be within the 
definite range R of either of the two adjacent sweeps. 
The chance for this is 2R/S = W/S when S > 2R 
= W, and unity when ^ IT. It is easy to see that 
the effective visibility E = W. 

The second case is that of the inverse cube law 
(which will be taken here in its simplest form). We 
obtain ^{x,S) with the aid of equation (26) 


^(a:,AS) = 2m 

n 


+ 00 

I 


(x —nSy 



csc^ 


ttx 


(44) 


the latter equality resulting from a well-known 
formula of analysis (obtained, e.g., from the expan- 
sion of the sine in an infinite product by taking 
logarithms and then differentiating twice). Inserting 
this expression into equation (43), we must find 

1^1 - exp csc2 

This is found by means of equation (38), on setting 
6 = TTx/S and X = 2mTr'^IS'^. The result, which can 
be transformed by means of equation (29), is 


P{S) = erf 




(45) 


These values substituted into equation (45) give 
F(S) = erf ^0.954 

A third case is useful to consider, although strictly 
speaking it is not one of parallel sweeps but of uni- 
form random search. It may be described as the 
situation which arises when the searcher attempts to 
cover the whole area uniformly by a path or paths 
which place about the same length of track in each 
strip but which operate within a given strip in the 


E\ 

s)- 


NS H 



manner of the searcher of Section 2.6. Let all the 
strips be cut by two horizontal lines a distance of h 
miles apart and suppose that the search is for a tar- 
get inside the large rectangle bounded by these lines 
and two vertical lines NS miles apart, as shown in 
Figure 11. The area is A = NSh square miles. As- 
sume that the total length of track is equal to that 
of all included parallel sweeps, L = Nb, then apply 
equation (40); we obtain 

P(S) = 1 - (48) 


We are now in a position to express m and W in 
terms of the effective visibility E. To find E we solve 


P(2E) = erf 


TT \/2m .a/ tt W 

2E ~ “W 


1 

2 


The tables of the probability integral show that erf 
0.477 = 0.5; hence 


\/2m Vi W 
"" 2E ~ 4E 


0.477, 


m = 0.046F2, W = 1.076F. (46) 



Figure 12. Probabilities with parallel sweeps. 

It is independent of N and h. 

These three cases may be represented by means 
of a common diagram (Figure 12) by plotting 




FORESTALLING 


31 


P = P{l/n) where n = 1/S is the sweep density, or 
number of sweeps per mile. At one extreme is the 
case of the definite range law, at the other the case 
of random search. All actual situations can be re- 
garded as leading to intermediate curves, i.e., lying 
in the shaded region. The inverse cube law is close 
to a middle case, a circumstance which indicates its 
frequent empirical use, even in cases where the special 
assumptions upon which its derivation was based are 
largely rejected. 


28 FORESTALLING 

When the observer is using two different means of 
detection simultaneously and independently (i.e., 
when neither interferes with or aids the other), it is 
sometimes necessary to know the probability of mak- 
ing a first detection by a particular one of the two 
means. Since the second means of detection can de- 
prive the first of a chance of detecting (by detecting 
the target first) this probability may be lower than 
if the second means had not been present: We say 
that the second can forestall the first. For example, 
when both radar and visual detection are possible, 
in gathering data bearing on the effectiveness of the 
radar, the possibility of visual forestalling of the 
radar must be taken into account. 

Again, when the target is itself capable of detect- 
ing the observer, and if it is important to detect the 
target before it can detect the observer (as when the 
former is a surfaced submarine which can submerge 
if it detects the observer first and so deprive it of its 
chance of detecting), it is important to find the prob- 
ability that the observer detect the target first, be- 
fore it has been forestalled by the target's detection. 

Just as the chance of detection is mathematically 
equivalent to that of hitting a target continuously 
exposed to our fire (intensity varying in general with 
the time), so the question of forestalling is mathe- 
matically identical with that of hitting the target 
before it hits us, in the case where it is an enemy 
continuously firing back. 

It will be sufficient to consider the case of con- 
tinuous looking with the instantaneous probability 
ytdt for the first means of detection without fore- 
stalling (Section 2.2), the probability p{t) of detec- 
tion when there is no forestalling being given by 
equation (6). Let y/dt and p'{t) be the corresponding 
quantities for the second means of detection, or for 
the target's detection of the observer in the second 
example above. 


If P{t) is the required probability of first detection 
during the interval of time from 0 to ^ by the first 
means, we consider the value of P{t dt). It is the 
probability of an event which can succeed in either 
of the following mutually exclusive ways: either by 
having the required detection between 0 and t, or 
by having neither means detect during this period, 
and having a detection by the first means between t 
and t dt. This leads to the equation 

P{t + dt) = P(t) + [1 - p{t)] [1 - p'(t)]yt dt, 

whence a differential equation is obtained, the solu- 
tion of which is 


Pit) 




(jt + ( 49 ) 


Precisely the same reasoning leads to the expression 
It' exp 

for the probability of a first detection by the second 
means in the time interval 0,^. 

Note that the sum Pit) + P'it) is the probability 
of a first detection either by the first means or by 
the second, in other words, the probability of a de- 
tection by some means between 0 and t. The expres- 
sion obtained by adding the above equations and 
carrying out one integration is 

Pit) + P'it) = 1 - exp (yt + yt)dt^, 

which is simply the expression (6) with jt replaced 
by Jt + y/, the latter being the instantaneous prob- 
ability when both means of detection act in con- 
junction (additivity of potentials). 

When a large number of independent trials of the 
detection experiment are made under identical con- 
ditions and all cases which have resulted in a first 
detection by the first means are sorted out and the 
precise epochs of this detection are averaged, the 
result will (statistically) be equal to 




iyt + yt')dt^ 

t = — 

£ Jt exp |^“J^ (yt + yt')dt^ 


dt 


dt 


(50) 


The denominator is proportional to the total number 
of first detections by first means, and the result of 




32 


TARGET DETECTION 


dividing jt exp 



(it + yt)dt 



the de- 


nominator is the proportion of such detections be- 
tween t and t + dt. Thus the expression in equation 
(50) represents the expected value t of t, the epoch 
of detection by the first means. 

As a first application we consider the case of con- 
stant instantaneous probabilities of detection, yt = 7, 
jt = t'. Equations (48) and (49) reduce to 


t 

7 + 7 


It is thus seen that the proportion of the total 
number of first contacts by the first means (as 
t—^co) is 7/(7 -f 7'), and, correspondingly, by the 
second, y'/{y + 7'). And the mean time elapsed 
to the former is the same as for the latter, i.e., 
1/(7 + y ')) this is different from the mean time I/7 
when no forestalling had been possible. 

As a second application we consider the straight 
track case of Section 2.4, and assume that the ob- 
server is an aircraft and the target a surfaced sub- 
marine. If the observer sights the wake of the sub- 
marine, his ability to detect may reasonably be taken 
as the inverse cube law of equation (10), and if the 
submarine sights the horizontal surfaces of the air- 
craft’s wing, the same law (with k replaced by a 
different constant k') can reasonably be assumed for 
the submarine’s detection of the aircraft. If it is 
assumed that the submarine dives as soon as it de- 
tects the aircraft, what is the probability that the 
aircraft detect the submarine, as a function of lateral 
range a;? By how much is its effective search width 
decreased by this new possibility? 

Equation (10) under the circumstances of Section 
2.4 leads to 


yt 


kh 

(x2 + 


k'h 

{x‘‘ + wH^/’ 


only in the present case the time interval is from 
— 00 to ^ instead of from 0 to t. With these changes 
equation (49) leads to the following expression for 
the probability of sighting the submarine before the 
time t: 


P(x,t) 


dy 


where m = kh/w and m' = k'h/w. The integral can 




be evaluated explicitly when it is noted that the 
integrand is proportional to the derivative of the 
exponential expression, i.e., 

^ exp f- (1 + 

= ^ exp T- (1 + 

m m dy [_ \ a/ x^-\-y^/J 

The result, on setting ^ = -j- co , gives the following 
probability of sighting the submarine some time on 
its whole straight course. 

P(X,00) ^ [1 - e-2(m + «')/*■]. 

m + m' 

To find the value of the search width (which will 
be denoted by W*), this expression must be used in 
the place of (26) in equation (27). The answer is 
obtained from (29) by replacing m by m + m' and 
then multiplying the result by m/{m + m'); it is 

\m-\- m \m-\- m 

Thus the effect of forestalling is to multiply IT by a 
factor less than unity of \/ml{m + m')- And the 
probability of detection even when the target is 
flown over (x = 0) is P(0,oo) = m/(m + m') instead 
of unity, as it would have been in the absence of fore- 
stalling. 

For a definite range law, that means of detection 
which has the greater range will always forestall the 
other. (Of course this is strictly true only in the case 
of continuous looking.) 


29 CONCLUSION— OPERATIONAL 

DISTRIBUTIONS 

Returning to first principles, as set forth in Sec- 
tions 2.1 and 2.2, it has been laid down as basic that 
detection, even when possible, is an uncertain event; 
and the whole subsequent course of development of 
this chapter has been toward the calculation of prob- 
abilities of detection. But an essential restriction has 
been imposed in all these calculations : The one source 
of uncertainty which has been considered is the 
human fallibility of the observer, and the sudden 
uncontrollable fluctuations in the physical state of 
affairs, but not in the random element introduced 
by unknown, long-term variations in the underlying 
physical conditions (conditions which are expressible 




CONCLUSION — OPERATIONAL DISTRIBUTIONS 


33 


as parameters). Thus, as we have said in Section 
2.1(ii), under given meteorological conditions of visi- 
bility V the observer will have a definite chance 
y(r)dt of sighting a target of given size A and back- 
ground contrast* C ; and subsequent deductions have 
been made on the assumption that while the range r 
may vary in a given manner in the course of time, 
the parameters F, A, and C all remain fixed. The 
distributions calculated on this assumption can be 
expected to agree with the distributions found em- 
pirically when the results of a large number of 
experiments are obtained, all of which are performed 
under the same conditions of visibility and size and 
contrast of the target, geometrical quantities like r 
alone being allowed to vary. But as soon as opera- 
tional results are compiled which refer to cases in 
which V, A, and C vary from incident to incident, 
an altogether different situation is present : The cause 
of the uncertainty of the event of detection is two- 
fold, being dependent not only on the human falli- 
bility of the observer and short-term fluctuations, 
but on the more or less unknown and heterogeneous 
nature of the underlying physical conditions. And 
it is important to realize that in many cases this 
second factor may outweigh the first. When this is 
judged to be the case, it may well be expedient to 
employ a highly simplified law of detection, such 
as the definite range law, and then seek to explain 
the distributions found in the operational data simply 
by averaging the calculated results of such laws over 
different possible values of the parameters. Thus if 
the definite range law is assumed, mathematical 
expressions deduced from it will involve this range 
R ; then it may be considered that in the operational 
incidents different values of R are present; by choos- 
ing appropriate frequencies for the different values 
of R and combining or averaging the theoretical re- 
sults over such distributions of 7^, a good agreement 
may often be found with the observations. 

It must be emphasized that equations such as (1), 
(3), (5), and (6) are true only when the first cause 
of uncertainty alone is present, and when the under- 
lying physical conditions remain constant (and are 
known to be of constant, though not necessarily of 
known values) throughout the course of the looking. 
Thus in proving (1), the probability of detection for 
one glimpse was g, of not detecting, I — g; now pre- 
cisely at the point where it was asserted that the 
probability of failure to detect at each and every one 
of the first n glimpses is (1 — ^)”, the assumption 
that the n different events are independent was made. 


This is justified only in two cases: first, when the 
only uncertainty is in the observer’s chance per- 
formance so that his different opportunities (glimpses) 
are regarded as repeated independent trials (as in 
successive tosses of a coin); second, when there are 
indeed changes in physical conditions, but of such a 
rapidly fluctuating character that if no detection is 
known to occur at one glimpse, no inference can be 
drawn regarding the physical conditions pertaining 
to any other glimpse. But if, for example, the visi- 
bility V is not fully known, the fact that earlier 
glimpses have failed to detect may lead to the pre- 
sumption that V is less than might otherwise have 
been supposed, and hence that later chances of de- 
tection are less: the expression (1 — g)” is false. 

The method of procedure is clear. The first step 
is to carry out the calculations as described in the 
previous sections of this chapter, assuming fixed con- 
ditions (such as V,A,C). The second step is to 
average the results obtained for the probabilities 
(e.g., over the possible values of V,A,C, with ap- 
propriate weighting). Only the final result can reason- 
ably be expected to furnish the probabilities which 
accord with the operational data. What is true of 
probabilities is also true of mean or expected values 
defined by them. 

This will be illustrated by many practical ex- 
amples, particularly in Chapters 4, 5, and 6. But three 
simple cases can be mentioned here. Firstly, suppose 
that the lateral range curve (Section 2.4) involves a 
parameter X referring to an unknown factor in the 
underlying physical conditions. Its equation is p = 
p{x,\). Once the distribution of frequencies with 
which the different values of X occur in an opera- 
tional situation has been estimated, the operational 
lateral range curve p = Pop(x) (i.e., the one furnished 
by a histogram of the observed data) is found by 
averaging pix,\) over the values of X on the basis 
of this frequency. Thus it might be reasonable in 
some cases to assume that the values of X are nor- 
mally distributed about a known mean Zwith a known 
standard deviation c. Accordingly, 

Popix) = — ^ 7 = f p(x,\) 

(7 V 27r J -00 

Thus if p(x,X) results from a definite range law of 
range R = \, so that 

p{x, X) = 1, when x < \, 

p{x, X) = 0, when x > K, 




34 


TARGET DETECTION 


the equation becomes 



the graph of which is shown in Figure 13. 



It is noted that when x = 0, Pop(b) is slightly less 
than unity, whereas it should exactly equal unity. 
This is because the normal distribution of definite 
ranges allows a (slight) chance of negative ranges, a 
physical absurdity. It would have been more realistic 
to have assumed a skew nonnegative distribution 
(e.g., Pearson’s Type III distribution, 

As a second example, consider the search for a 
fixed target by two parallel sweeps at distance S 
apart. If the underlying conditions are the same 
during the two sweeps, and if p(x) is the lateral range 
probability, the chance of detection of a target be- 
tween the paths and x miles from one of them is 
shown by the usual reasoning to be 

P{x,S) = 1 - [1 - p{x)] [1 - piS - a:)] 

= + P{S - x) - p{x)p{S - x). 

If p{x) = p{x,\), a weighted averaging process 
must be performed in order to get the operational 
probability Pop{x,S) from P{x,S,\) given by the 
above equation. And of course if x is determined at 
random between 0 and S, a second averaging must 
be done to get Pop{S), the operational probability 
of detecting the target given only to be somewhere 
between the sweeps and with a given distribution of 
physical conditions. The order in which these two 
averagings are done is immaterial. A corresponding 
treatment is given in the case of infinitely many 
parallel sweeps. It may be remarked that the opera- 
tional effective visibility which is defined by the 
equation (see Section 2.7): 


P„p(2jB„p) = J P(2P„p,X)/(X)(iX = I 

is quite different from the average E : 

E = j E(\)f{\)dK, 

• 

of the effective visibility defined under fixed condi- 
tions corresponding to a particular value of X. Here 
/(X) is the assumed frequency with which the values 
of X are taken to be distributed under the operational 
conditions in question. 

As a third example, suppose that a radar set is 
chosen at random from a lot, only the fraction e of 
which are in good adjustment, the remaining 1 — e 
not in a condition to make any detections possible. 
The radar lateral range curve p{x) for a radar set in 
good adjustment and, e.g., mounted on an aircraft, 
must be multiplied by e to obtain the operational 
curve that will be obtained when many observations 
are made with the aid of many sets chosen in this 
way. When a set or similar observing instrumentality 
or setup is not giving the results which could be ex- 
pected of it, it is often said to be working at an 
efficiency less than 100 per cent. In the above case, 
a natural definition of efficiency is lOOe. In more 
comphcated cases, the concept, while useful as a 
general concept, may not be convenient to define in 
all precision. 

In conclusion, the following principle is laid down : 

If the object of the calculation of probabilities, aver- 
ages, and similar statistical detection quantities is to 
coordinate and explain the data of the operations of the 
past, then the heterogeneity of conditions {dispersion of 
slowly varying parameters) is placed at the apex of the 
discussion, the influence of “subjective” probabilities 
and short-term fluctuations {the main subject of this 
chapter) usually playing a secondary role. 

If on the other hand the object of the calculation is 
to obtain contemplated performance data for the design 
of search plans to be used in the future, and, as is gener- 
ally the case, when the conditions {slowly varying param- 
eters) are known, then the probabilities originating 
from subjective and rapidly fluctuating sources occupy 
the center of the stage; any study of the heterogeneity of 
conditions is made only in order to check the sensitivity 
of the search plan to accidental imperfections in the 
knowledge of the conditions. 

All this will be made clear on the basis of examples 
in the succeeding chapters. 


Chapter 3 

THE DISTRIBUTION OF SEARCHING EFFORT 


3.1 


THE GENERAL QUESTION 


I N AN IMPORTANT CLASS of problems of naval search, 
the target has an unknown position but a known 
distribution: Avhile we do not know where it is, we 
do know the probability that it is in this place rather 
than in that. If, as is usual, the total available search- 
ing effort (number of hours the observer can devote 
to the search) is limited, how should the searching 
be done? How much time should the observer spend 
searching in this place and how much in that? 

In other cases, there is no doubt as to where to 
search but there may be a question as to when. If 
the target is only temporarily present in the region 
where the search can be made, all hours of the 
twenty-four being equally likely, should the limited 
searching effort be spread out evenly during the 
twenty-four hours, or should a more intensive search 
be conducted during only part of the time? This 
question is of particular interest if the search is done 
by aircraft, when only a limited number of aircraft- 
hours are available, and if search in daylight when 
vision and radar can be used simultaneously is more 
effective than search at night when only radar is 
available. How should the search be divided between 
day and night? 

There are similar questions concerning the opti- 
mum distribution of scanning effort : An observer on 
an aircraft is searching for a surface target; how 
much of his time should be devoted to looking 
straight ahead, how much in looking abeam, and 
how much on the intermediate bearings? 

There is a close mathematical analogy between 
some of these problems and certain questions of 
gunnery and bombing in which the distribution of 
targets is known, and the optimum distribution of 
firing is required. 

It has been seen in Chapter 2 that when a region 
is searched the chance of detection depends not only 
on the law of detection (lateral range curve, search 
width, etc.), but on the method of search. At one 
extreme, such a highly systematic method as that 
of parallel sweeps can be used; at the other extreme, 
random search can be employed, leading to the equa- 
tion (40) of Chapter 2, 

p = 1 - (1) 



where 

A = area searched (square mi 
W = effective search width (miles), 

L = length of observer’s path inside A (miles), 
p = probability of detection of a target given to 
be in A and uniformly distributed therein. 
Throughout this chapter, the problems just men- 
tioned will be treated on the basis of equation (1). 
The reason for the assumption of random search is 
twofold. On the one hand it is realistic, since in any 
protracted search, however systematic in intent, 
navigational errors and other irregularities and un- 
certainties are pretty sure to impart to the search a 
character of random; hence (1) is a proper estimate, 
on the conservative side, of the practical results 
achieved. On the other hand, the assumption is con- 
venient and leads to usefully simple results; this is 
partly because equation (1) requires nothing con- 
cerning the particular detection law (other than the 
value of W) to be assumed, and partly because of the 
usable nature of (1) itself. 


32 ALTERNATIVE REGIONS OF SEARCH 

Let Ai and A 2 be two areas of the ocean (either 
separate or having a common boundary). The target 
to be found is either in A 1 or in A 2 , with the respective 
probabilities pi and p 2 of so being, so that relations 

Pl + P2 = 1, Pi > 0, P2 > 0 

hold; the target being stationary, pi and p 2 do not 
change with the time. Let the target be uniformly 
distributed in whichever of A 1 or A 2 it lies. Finally, 
let the total length of track of the observer (or ob- 
servers) be L. How must L be distributed between 
Ai and A 2 if the chance of detection is to be greatest? 
In other words, if L = Li -j- L 2 , Li being the length 
of the observer’s track in Ai, L 2 that in A 2 , what re- 
lation must exist between Li and L 2 for the optimum 
search? If W is the search width, the probability p 
of detection is given by 

p = Pi (1 - + P2 (1 - (2) 

as results from a simple probability argument based 
on (1). 


■rnNKIXlEMXIAfc 


35 


36 


THE DISTRIBUTION OF SEARCHING EFFORT 


Mathematically, the problem is to find the values 
of Li and which maximize equation ( 2 ), subject to 
the conditions 

Li L2 = L, Li ^ 0 , L2 ^ 0 . ( 3 ) 

It is convenient to proceed graphically. Setting 


Then the inchnation of yi ^\i]l continue to be still 
less than that of 2/2 throughout the interval: No 
internal minimum of y exists, but since 2/2 increases 
faster than 2/1 decreases as x moves from 0 to L, 
the minimum occurs at x = 0 (i.e., Li = 0 ). 

Case 2 . The inchnation of 2/2 at x = L is less than 
(or equal to) that of 2/1 at this point, i.e.. 


Li — X, Z/2 — L — X, 
Wx/Ai 


2/1 = Pi € 
we have 


2/2 

p = 1 ■ 


-W{L-x)/Ai 


92 6 

(yi + yt), 


so that, for the optimum search, x must be deter- 
mined so as to maximize p, i.e., to minimize 


^ ^ ^-WLlAi 

A2 Ai 


( 5 ) 


Then, by a similar argument, the minimum occurs at 
X = L. 

Case 3 . Neither one of the above cases occurs: 


2 / = 2/1 + 2/2, 

subject to the restriction that 0 ^ x ^ L. Figure 1 
shows the graph of y against x in a typical case ; the 
ordinate is obtained by adding the ordinates of the 



graphs of 2/1 and 2/2, also shown in the figure; the 
latter are simple exponential curves. It is seen, either 
by differentiation or by an obvious graphical argu- 
ment, that for y to have a minimum at a point x in 
the interval ( 0 , L) it is necessary and sufficient that 
the inclinations (\dz., absolute values of the slopes) 
of the tangents to the 2/1 and the 2/2 curves be equal 
and opposite (they are always opposite in the present 
case). Now the inchnation of the 2/1 is always down 
and that of the 2/2 always up. The former inchnation 
decreases with increasing x from its maximum at 
X = 0 , the latter increases to its maximum at x = L. 
Thus there are three mutuahy exclusive possibihties. 

Case 1 . The inchnation of 2/1 at x = 0 is less than 
(or at most equal to) that of 2/2 at this point, i.e., 

A\ A2 


T> 

Ai 


P2 -W 


and ( 6 ) 


Then the inchnation of 2/1 at x = 0 is greater than 
that of 2/2 at this point, but this advantage is steadily 
diminished as x increases, and is reversed ^vhen 
X = L, at which point the inchnation of 2/2 exceeds 
that of 2/1- Hence there is just one point x — Xo in 
the interval ( 0 , L) which makes y a minimum. This 
value of X is found by sohdng the equation dy / dx = 0 ; 
a convenient form of the answ^er wdU be given below. 

In order to grasp the meaning of the situation more 
easily, the following terms are introduced: 

Pi = Pi/^i, P 2 = 92/ A2 

<t>i = WLi/Ai, <t>2 = WL2/A2 ( 7 ) 

A = Ai-h A2, ^ = WL. 

On account of the uniformity of the distribution of the 
target in w^hichever region it is, the probabihty that 
it he in a subregion of unit area within Ai is the 
product of the probabihty pi that it be in A 1 by the 
probabihty that (in this case) it be in said subregion, 
1 /Ai: Thus the probabihty in question is pi/Ai = pi. 
Hence pi is the probability density for the first region; 
P2 has the corresponding meaning for A2. Further- 
more, the length of observer's path Li in Ai is a 
measure of the searching effort devoted to Ai. But 
TFLi is an equally good measure: it is the area swept 
(some of it multiply) in Ai. Thus the expression 
TFLi/Ai = <f>i is the density of searching effort in the 
first region, and <t>2 is that in the second. <I> is the total 
available searching effort, and w-^e have 

Ai<t>i + A24>2 = </>i = 0, 4>2 = (8) 




ALTERNATIVE REGIONS OF SEARCH 


37 


In the third case, corresponding with equation (6), 
the solution of dyjdx = 0 for a; gives a result which 
can be simplified by first replacing x and L — xhy 
Li and L 2 , and then replacing ratios by the quantities 
introduced in equation (7). The following equations 
result from this process; they answer the question 
concerning the optimum distribution of searching 
effort in the present case. 

1 $ 

01 = log Pi - j (^1 log Pi + A2 log P2) + -j 

1 <I> 

02 = log P2 “ ^ (^1 log Pi + A 2 log P 2 ) + ^ • 

On the basis of equations (4), (5), and (6), ex- 
pressed in the terms of equations (7), and (9), every- 
thing may be summed up as follows : 

When the target’s probability density pi in the 
first region is not only less than its density in the 
second, but is so much less that it remains less when 
the second is multiplied by the factor (< l)^ 

i.e., 

( 10 ) 

then no searching whatsoever should he done in the first 
region, Ai, and the whole effort should he devoted to 
searching the second, A 2 . Similarly, if 

P2 g Pie ( 11) 

no searching should be done in A 2 . When, on the 
other hand, 

Pi>P 2e-*^^’ and P2>Ple'*^^ (12) 

the searching effort should be distributed in accord- 
ance with equation (9) . If k denotes the exponential 
of the common value added to log pi or to log p 2 in 
(9), these equations become 

01 = log kpi, 02 = log hp 2 . (13) 

Thus the optimum densities of searching effort [in case 
of equation (12) ] equal the logarithms of quantities 
proportional to the respective prohahility densities. 

It is interesting to see how the situation develops 
as the total available searching effort 4> is progres- 
sively increased. To have a definite case, suppose that 
Pi > P2 and Ai < A 2 . When 4> is very small, equation 
(11) holds: all searching must be done in Ai, none 


in Ao. When 4> increases sufficiently, (12) becomes 
valid, and remains so for all further increase in <1>. 
Then the searching has to be distributed between Ai 
and A 2 according to the logarithmic law enunciated 
in equations (9) or (13). This leads to a curious con- 
clusion as 4> becomes extremely large. For equation 
(9) shows that 

lim ^ = 1. 

Thus for very large 4>, 0i is about equal to 02, i.e., 
(9) WLi/Ai ^ WL 2 IA 2 , which means in the present case 
where Ai < A 2 that Li < L 2 . The first region, which 
for very small $ should take up all the searching 
effort, should for very large 4> actually have a shorter 
length of observer’s track than the second region, a 
phenomenon of reversal for large 4>. 

Suppose that after completing the optimum search 
with the expenditure of the total effort 4> = TFL, with 
no resulting detection, a further amount of effort 
4>' = WL' becomes available. What is the optimum 
manner of expending this additional effort? 

Assume that in the first part of the search, the 
third case was presented, so that was distributed 
in accordance with equation (9). Since, as we are 
assuming, the target is not detected, the situation 
at the end of the search is similar to that at the be- 
ginning, except that the probabilities Pi and p^ have 
to be replaced by different values pf and pf, the 
values of which are computed by means of Bayes’ 
theorem (see Section 1.5, in particular footnote c). 
In this application, the a priori probabilities of Ai 
and A2 containing the target are Pi and p^, the a 
posteriori probabilities are pf and pf. The ‘‘pro- 
ductive probabilities,” i.e., those of not finding the 
target in Ai or in A 2 when the 4> search is done as 
assumed in (9) are the values which accrue to the 
quantities and when Li and L 2 are 

given by (9) in conjunction with (7). Thus the first 
productive probability is, by definition, the chance 
that if the target is actually in Ai the searching effort 
01 = WLi/Ai devoted to this region shall fail to 
reveal it; by the formula of random search, this is 
e~WLiiAi^ and similarly for the second region. Hence 
by Bayes’ formula. 

“ pi + p2 ’ ’’ 

or, using equations analogous to (7), 

Pi 

Ai Pi e~'^^ + A 2 P2 ’ 


f = 1 or 2. 


38 


THE DISTRIBUTION OF SEARCHING EFFORT 


To find the optimum distribution of observe 
first that on account of (9), the relation = 

P2e“'^' holds; hence we are in the presence of the 
third case [the primed analogue of (12) ]. Hence 4>\ 
and 4>2 are given by equations like (9) (with appro- 
priate primes). An obvious algebraic simplification, 
using the original (9), shows that 

Cf>l' = <i>2' = -• 

A 

Now the total density of searching effort devoted 
to the ith region (?; = 1 or 2) is simply + 0/. This 
reduces with the aid of (9) to 

I $ + <!>' 

log Pi ~ ^ (^1 Pi + ^2 log P2) + — — • 

But this is exactly what (9) would have given if we 
had known in advance that the total amount of 
searching effort would be ^> + $' rather than ^>. In 
other words: 

A well-planned search can not he improved hy a re- 
distribution of search made at an intermediate stage of 
the operation in an attempt to make use of the fact that 
up to that time the target had not yet been observed. 

Of course as soon as the target is observed, an im- 
provement can be made: Discontinue the search. 

This theorem has been proved here only in the 
case where equation (12) is valid. Other cases are 
treated in a similar manner, with similar results. 

In the case of equation (12), formula (9) gives only 
the magnitude of effort to be devoted to A\ and to A2; 
it does not tell how the search should be conducted 
in time. Suppose that at the end of t hours the 
amount of search effort is <I>(Q = d, where <I> = 4>(T) 
= cT, T being the total time available and c a 
constant of proportionality. Then in order to find the 
target as soon as possible, we must proceed as fol- 
lows : Whatever the value of t (> 0), the search effort 
ct must be used so as to maximize the chance of 
detection up to that time, in accordance with the 
formulas developed above. Thus, if pi > p2, we must 
make <^i = ctJAi, <^2 = 0, as ^ goes from zero to 
A 2 (logp2 — log pi)/C, i.e., when (12) comes into effect. 
From then on, <^i and <^2 must be taken from equation 
(9) in which <I> is replaced by ct. 

An obvious extension of the problem treated in 
this section is to the case of n regions Ai, • • *, An. 
But, while this presents no difficulty, it is much more 
worth while to treat a perfectly general case which 
will yield the n region case by specialization. This 
will be the object of the succeeding section. 


33 the general case— targets 

CONTINUOUSLY DISTRIBUTED 

The target is stationary and is contained in a 
known region A of the ocean (assumed to be a plane 
in which a cartesian system of coordinates {x,y) is 
established). The probability before the search that 
the target be in the infinitesimal region dxdy is 
p(x,y)dxdy, the probability density p{x,y) being 
known and satisfying the conditions of continuity, 
of having in A the positive minimum. 

min p{x,y) = po > 0, 

A 

and of satisfying the obvious equation. 

p(x,y)dxdy = 1. 

A 

The total available searching effort as represented 
either by the total observer’s track L or by ‘F = WL 
is given. How shall the searching be distributed 
throughout A in order that the chance of finding 
the target be a maximum? 

The idea of a distribution of searching effort in 
the present case of a continuously varying distribu- 
tion involves a less precise conception than in the 
case of Section 3.2. It is necessary to arrive at the 
notion of density of searching effort (t>(x,y) as a 
function of position (x,y) in A. This is accomplished 
as follows : 

Consider a subregion B within A. Let Lb be the 
total length of observer’s track in B (composed, per- 
haps, of many pieces). The quantity <^5 = WLb/B 
obviously corresponds to B in the same way in which 
</>! of (7) does to Ai in Section 3.2. But consider what 
happens as B shrinks up to fixed point {x,y) con- 
tained therein. If B is large and of not too irregular 
a shape [if it is square or circular with {x,y) at its 
center], the ratio Lb/B behaves as the ratio of the 
number of molecules of an inhomogeneous body in a 
volume to the volume itself as it shrinks: After 
changing slowly, it settles down to a quasilimiting 
value, remains at this value for a long time, only to 
depart radically from it as the area (or volume) falls 
below a critically small size. The quasilimit I {x,y) of 
Lb/B (corresponding to the “mean number of mole- 
cules per unit volume,” proportional to the density 
of the body) may be called the density of observer 
track at {x,y). The product <h{x,y) = Wl{x,y) = 
quasilimit of WLb/B shall be called the density of 
searching effort or search density at ix,y). An obvious 
construction shows that 





THE GENERAL CASE — TARGETS CONTINUOUSLY DISTRIBUTED 


39 


l{x,y)dxdy = L, 

A 

and consequently that 

J J'<^(x,y)dxdy = WL = <!>. (14) 

A 

Suppose that the search density (i>{x,y) is a given 
function. What is the probability P[4>] of detecting 
the target? The classical reasoning of the integral 
calculus (subdivision of the region A ; approximation 
to P[<^] by a sum obtained by total and compound 
probability; and apphcation of the formula (1) in 
the limit) furnishes the following answer: 

P[4>] = ff P(^,y)(l - e~‘^^'''^^)dxdy. (15) 

A 

We are therefore in the presence of a problem of 
the calculus of variations: Among all functions (f>{x,y) 
which satisfy (14) together with 


whereupon the following solution of the problem is 
obtained from (17) : 

<t>{x,y) = log p{x,y) - jj f\ogp{x,y}dxdy + j- 
^ (19) 

Returning to the original problem with (16) in 
force, it is seen that equation (19) automatically 
gives the solution whenever <I> is large enough to make 
the right-hand member of (19) nonnegative for all 
(x,y) of A, i.e., whenever (po being the minimum of 
p(x,y) in A) 

log Po - jff log p{x,y)dxdy + 2 ^ 0 . (20) 

A 

In case this inequality is not realized, we shall allow 
the previous formulas to suggest a (f>{x,y) which is 
written down ad hoc, and then show a posteriori 
that it is in effect the solution of the problem. 

Introduce the variable q ranging in the interval 



<l>(x,y) S 0, (16) 

find that one which gives to P[(f)], defined hy (15), the 
greatest value. 

This is not a ‘Tegular” problem of the calculus of 
variations, because of the one-sided condition (16), 
i.e., because this condition is an inequality rather 
than an equality. And indeed the example of Section 
3.2 prepares us for the possibility that under certain 
conditions the maximum cannot be found by simply 
equating a combination of differentials (variations) 
to zero. 

As a preliminary step, we shall solve the regular 
problem similar to the above but with the condition 
(16) omitted. The use of the Lagrange multiplier X 
in the variation of (14) and (15) leads to 

b j'J*p(x,y)(l — e~'^^'^'^^)dxdy — X5 ^ ^4>{x,y)dxdy = 0, 

A A 

f f - X] 8</)(x,y)dxdy = 0, 

A 

from which is derived 

p(x,y) = X, 

4>{x,y) = log p{x,y) - log X. 

To determine X, introduce this in (14) and solve for 
logX; 

logX = 2 J f log p(x,y)dxdy - p (18) 

A 


PqS q ^P\ = max p{x,y), 

_ A 

and denote by Aq and Aq the two parts of A (i.e., 
A = Aq-\- Aq) defined as follows: 

p(x,y) ^ y for all (x,y) in Aq, 
p(x,y) < q for all {x,y) in Aq. 

Consider the quantity 


( 21 ) 


//' 


[log p(x,y) - log q\dxdy 


= -A, 


log q + p{x,y)dxdy. 


Geometrically, it represents the volume of that part 
of the solid under the surface z — log p{x,y) [plotted 
in the cartesian coordinates {x,y,z)] cut off by the 
plane z — log q (and above this plane). As q de- 
creases continuously from pi to po, this volume in- 
creases continuously from 0 to 

-A log Po+ff log p(x,y)dxdy. 


Hence the expression 

Aq log q - JJ log p{x,y)dxdy 




decreases continuously from 4> to the value 
A log Po — J" J log p{x,y)dxdy + <I> 




40 


THE DISTRIBUTION OF SEARCHING EFFORT 


which, on account of the assumed invalidity of (20), 
is negative. There exists, therefore, a unique value 
q = h for which the expression is zero, i.e., for which 


log 6 


-U/' 


$ 


log p{x,y)dxdy + — = 0. (22) 


We now define <i>{x,y) as follows: 

<l>{x,y) = log p{x,y) - A j J\ogp{x,y)dxdy + -^ 


= log 


p(x,y) 


b 

(23) 


when {x,y) is in 

(f>(x,y) = 0 

when (x,y) is in 


To show that this 4>{x,y) is the solution of the 
problem, it is observed, firstly, that it satisfies (16) 
in virtue of (22) and (21); and, secondly, that (by 
direct integration) it satisfies (14). It remains to 
show that of all the functions satisfying these two 
conditions, it renders P[(f)] a maximum, for which we 
will regard it as sufficient to show that every ar- 
bitrarily small change of <t>(x,y) through values satis- 
fying (16) and (14) decreases P[</)] — or leaves it 
stationary. 

First, consider variations which leave the function 
zero in Ab, i.e., which correspond to a rearrangement 
of values inside A^. To find the maximum, repeat 
the preceding calculation starting from the equation 
dP — X54> = 0 with A^ replacing A; this leads to 
(19) with A replaced by Ab, i.e., to (23). 

Second, consider variations which transfer some of 
the searching effort in Ab to Ab. They are obtained 
as follows: Let yp{x,y) be any function continuous in 
A and satisfying the conditions 


^{x,y) ^ 0 for {x,y) in Ab, 
^{x,y) ^ 0 for {x,y) in Ab, 


4>{x,y) + yl/{x,y) ^ 0 for {x,y) in Ab, 
J* j" ^{x,y)dxdy = 0. 


Evidently if 0 ^ ^ ^ 1, ^ ^ (a:, ?/) will have these same 
general properties as yp(x,y). Then <f>(x,y) + ^rp{x,y) 
represents the result of varying (t>{x,y) in the manner 
described. It remains to show that if 


P{i) = J'J* V{x,y) [\-e + ^'^^]dxdy 

A 


then P'(^) ^ 0. We compute as follows: 

P'(0) = V{x,y) xf/{x,y)dxdy 

A 

= ff v{^,y) \l/(x,y)dxdy 

+ JJ P(x,y) xPix,y)dxdy. 

^b 

From (23), 

p(x,y) = h when (x,y) is in Ab, 

p{x,y) < h when {x,y) is in Ab, 


hence 
P 


'(0) = 4'{x,y)dxdy + J" J'p(x,y)t/^(x,y)dxdy, 

^b Ab 

^‘[ff \P(x,y)dxdy + JJ 4'{x,y)dxdy~^ ’ 


^6 


b JJ \f/{x,y)dxdy = 0. 


A 

This completes the proof that (23) gives the solution 
required. 

It is possible to put this result in a geometrical 
form. In the rectangular coordinates of the variables 
x,y,z, plot the surface z = log p{x,y) over the region 
A in the xy plane. Cut this surface by the horizontal 
plane z = log h, choosing the constant b so that the 
volume cut off (above the plane and below the sur- 
face) shall equal 4>, and denote the orthogonal pro- 
jection upon the xy plane of the portion of the surface 
above the plane by A^. This construction is the geo- 
metrical counterpart of equation (22) defining A^. 
The search density ^(x,y) is then taken as zero out- 
side Ab (i.e., wherever the surface is below the plane) 
and equal to the length cut from a vertical line by 
the surface and the plane, at each point (x,y) of Ab 
(through which point the vertical line is drawn). 
This is a consequence of (f>(x,y) = log p{x,y) — log h, 
which is contained in (23). 

This construction shows how to lay on additional 
searching in case an additional search effort A4> be- 
comes available after <I> has been used up fruitlessly: 
Lower the horizontal plane so that the additional 
volume between it and the surface is A4>; then search 
throughout the (generally larger) region A^, with a 
search density equal to the length of the vertical 
segment between the new and the old horizontal 




AN APPLICATION 


41 


planes, or, in new parts of Af,, between the new plane 
and the surface. The resulting total layout of search 
density is obviously that which corresponds to the 
total available effort $ + A$. To prove that the 
knowledge, that the first part of the search (with $) 
has failed to detect the target, leads by Bayes’ 
theorem to the above as the optimum procedure, one 
employs the method of reasoning illustrated in the 
corresponding question in Section 3.2. 

These considerations show how to carry out a 
search in time as the available effort <I> = ^{t) in- 
creases progressively, the purpose being to detect as 
early as possible : Lower the horizontal plane z = log 
h at such a rate that the volume between it and the 
surface z = log p{x,y) constantly equals and 
constantly add search density by amounts equal to 
the additional lengths of vertical segments described 
above. 

It may be noted that a very slight change in word- 
ing of certain of the proofs in this section shows that 
the solutions here given continue to be valid when 
the condition po > 0 and pi finite are both dropped; 
and also, when the region A is infinite in area. 
Actually, p(x,y) need not even be continuous. And 
of course the distribution can be in a line or in space 
rather than in the xy plane, with appropriate reword- 
ing. Finally, this problem specializes and becomes 
like the one of Section 3.2, as is seen on setting 
A = Ai + 4.2 and taking p{x,y) equal to pi/Ai or 
P 2 /A 2 according as {x,y) is in Ai or in 4.2. Then 
equation (23) will reduce to such equations as (9), 
etc. Similarly, equation (23) yields the solution of 
the n region case spoken of in the last paragraph of 
Section 3.2. 

Corollary: If in all the previous discussion and 
equations, p(x,y) is interpreted not as the probability 
density of the distribution of an individual target in 
4, but as the expected number of targets per unit 
area, i.e., p(x,y)dxdy is now the mean or expected 
number in the infinitesimal dxdy region at (x,y) where 
there is an indefinite number or swarm of targets in 
4 (which may be an infinite region), and where the 
quantity to be maximized is not the probability of 
detecting the target but the expected number de- 
tected — then all the conclusions, equations, and 
geometrical constructions remain valid. 

This can of course be proved by paralleling the 
whole of the previous discussion. But it is simpler to 
observe that the probability of an arbitrarily chosen 
individual target of the swarm being in dxdy is 
proportional to p{x,y), and then to show that the 


constant of proportionality cancels out of the crucial 
equations (22) and (23) and makes no difference in 
Ajy, properly defined. 


AN APPLICATION 


Let an approximately stationary target be placed 
on the ocean according to a bivariate circular normal 
distribution centered at the origin of the coordinate 
system, and of standard deviation a; thus 


p{x,y) = r^- = + y^. (24) 

The region 4 of Section 3.3 is now the whole plane 
and Pq = 0; but, as has been remarked, the solution 
given in 3.3 is valid for such a case, and the optimum 
distribution of the available searching effort is 
given by (23) together with (22). 

Let a be the distance r at which p{x,y) assumes the 
critical value 5. Then 4^, is the region for which 
r ^ a, and evidently 

A, = wa\ h = ~ (25) 


Substituting these expressions into equation (22) and 
eliminating b, we derive the expression for a, 


a^ = 


4 ^ 

TT 


(26) 


Thus a is proportional to \/ a; also to $. 

Outside the circle centered at the origin and of 
radius a no searching should be done. Inside the 
circle the searching effort should be distributed ac- 
cording to the formula (23) which in this case reduces 
to 


4>{x,y) = 


a^ — r2 


(27) 


The graph of the equation z = <t>{x,y) in the space 
of the coordinates {x,y,z) is a paraboloid of vertex at 
(0,0,a^/2a-^), axis coincident with the z axis, cutting 
the xy plane in the above circle of radius a, and is 
the xy plane itself outside this circle. 

To consider a concrete case, let o- = 100 miles 
(which means that there is half a chance that the 
target is within a circle of 118-mile radius). Assume 
that there are five 130-knot aircraft, each available 
for five hours of search, so that L = 3,250 miles. 
Assume a search width on the present type of target 
of IF = 5 miles. Thus 4> = IFL = 16,250 square 


42 


THE DISTRIBUTION OF SEARCHING EFFORT 


miles; by (26) a = 121 miles. Only within a circle 
of this radius should searching take place, and there 
the path length per square mile should depend on the 
distance r from the center according to the formula 

<j> 1212 _ y.2 

W ~ 100,000 * 

3 5 DISTRIBUTION OF EFFORT IN TIME 

In certain tactical situations involving a planned 
search there is a question not of the distribution of 
searching effort in space but in time. In order to 
illustrate such cases, a typical but simple problem 
will now be considered. 

A certain relatively narrow region A of the ocean 
has to be crossed by very fast enemy surface units. 
It takes each one a definite time T to cross A, T 
being of the order of an hour or two. We wish to 
detect such units by means of aircraft patrolling the 
region A : Certain features of the tactical situations 
require them to stay within A at all times. We have 
at our disposal a fair number of aircraft of the same 
type, each capable of a definite number of flying 
hours during the twenty-four, 6 to 12, for example. 
Thus the total length of track of all aircraft during 
twenty-four hours has a fixed value of L miles; this 
is a measure of the total available searching effort. 
During the twelve daylight hours, radar search can 
be supplemented by visual, the combined power of 
detection being expressed by the value Wi of the 
search width for each separate aircraft. During the 
twelve hours of darkness, radar is the sole means of 
detection, and the search width falls to a value TF 2 , 

Wi > W 2 . (28) 

One final assumption is fundamental: The enemy, 
not being aware of our search of the region A, is as 
likely to cross this region at any one time as at any 
other. 

It may be assumed that for the best search the 
number of aircraft patrolling during any one hour of 
the twelve daylight hours should be the same as 
during any other. For if during a particular hour 
there are fewer patrols than during another, the loss 
of chance of detection during the former is not quite 
compensated by the additional chance during the 
latter, on account of the tendency of overlapping 
(saturation effect), which is always present, but in- 
creases with increasing number of patrols in a given 
region. This situation could of course be given a pre- 


cise mathematical formulation and proof, with which, 
however, we will dispense. In conclusion, we may say 
that in any scheme of search to be considered there 
exists a constant number ui of planes airborne dur- 
ing the day; and, by corresponding reasoning, a con- 
stant number, 712 , during the night. If v is the com- 
mon aircraft speed, the total daytime and nighttime 
length of track flown is I2vni and I 2 vn 2 miles re- 
spectively. Thus the following equation, expressing 
the limited total searching effort, must hold. 

= il;- 

Beyond this, Ui and 712 are at our disposal; they char- 
acterize the distribution of searching effort between 
day and night. 

If the enemy unit crosses A during daylight, the 
total length of track flown while he is in A will be 
Tvui (there are Ui planes in the air, each of speed v, 
during the time T of passage of the enemy). Hence 
by the formula of random search^ the probability 
of detecting such a target is 

^ ^ — Tvn\W\l A 

while the probability of detecting a target passing 
at night is 

Since the chance of the enemy’s crossing A by day 
or by night are the same, the sum of one-half of 
each of the above expressions gives the required 
probability of detection p (assuming for simplicity, 
and with satisfactory approximation, that the chance 
of a passage of A partly at night and partly in day- 
time is of negligible probability) : 

p = 1 — i [q-Txu^WxIA _|_ ^-TvmW 2 lAy (39) 

Mathematically, our problem is to choose ni and 
n 2 subject to (29) so as to maximize p. The details 
of the work are altogether similar to those of Section 
3.2, except that ni and rh are the variables in the 
present case. The results are as follows: 

Case 1. L is so small that 

g (31) 

alt is assumed here, as throughout this chapter, that the 
flights are at random. When the direction of the enemy target 
across A is known, a more efficient disposition of aircraft 
tracks is in the form of a crossover barrier patrol (Chapter 7) ; 
the formulas, hut not the ideas and essential results, will then 
be changed. 




OPTIMUM SCANNING 


43 


then 


ui = LjVlv, 712 = 0, (32) 

and all the flying must be done in daytime. 

Case 2. L is large enough to make 

(33) 

then both day and night searching must be done 
according to the equations [analogous to (9)] 




where i = 1 and 2. 


As long as L is only moderately greater than the 
critical size expressed in equations (31) and (33), 
i.e., for which (31) becomes an equality, more search- 
ing should be devoted to day than to night (ni > rh) . 
But in the limit, for increasing L, (34) shows that 
ni/n 2 approaches W 2 /W 1 , which is less than unity; 
thus again we have the phenomenon of reversal 
signalized in Section 3.2: If a very large amount of 
searching effort is available, more searching should 
be done during the unfavorable period than during 
the favorable one. 


36 OPTIMUM SCANNING 

Returning to the framework of ideas and notation 
of Chapter 2, suppose that the observer and target 
are on straight courses at fixed speeds, so that relative 
to the observer the target is moving in a straight line 
at the speed of w knots, his lateral range being x 
miles (see Chapter 1, Figure 5). It is convenient here 
to regard the target as moving down the line parallel 
to the axis of ordinates cutting the axis of abscissas 
at the point of abscissa x > 0. His position at the 
epoch t is at the point of coordinates {x,y), where x 
remains constant and y = —wt (the negative sign, 
because of his downward motion: y decreases as t 
increases). The observer remains at the origin. 

Instead of taking the instantaneous probability of 


detection ytdt for granted, as we largely did in 
Chapter 2, we here propose to inquire into it more 
deeply, and in particular to examine the effect of 
varying the method of directing the line of sight 
(or the radar or sonar beam) from position to position 
over the field of view. We propose, in other words, to 
examine the effect of different scanning procedures 
upon 7f, and through it, upon the search width W. 
And we will say that the scanning method is optimum 
if it renders W a maximum. 

For any method of detection (visual, radar, sonar, 
etc.), there exists a quantity X defined as follows: Let 
the relative positions of the target and observer re- 
main virtually unchanged during a short interval of 
time dt, during which the observer directs his axis 
of vision (or the radar or sonar beam axis) straight at 
the target. Assuming no previous detection, the prob- 
ability of detection during dt is \dt, which can be 
called the instantaneous line-of-sight detection prob- 
ability. We shall assume that X depends only on 
range, X = X(r). We shall assume furthermore that 
the probability of detection during dt is insignificantly 
changed by a slight change in the position of the 
target out of the line of sight, e.g., by an angle less 
than e radians, but that it falls virtually to zero at 
greater angles. The reader will appreciate that this 
assumption is not unrealistic in the important case 
of a target close to the threshold of visibility (or with 
narrow radar or sonar lobes). 

For any method of scanning, there exists a function 
f(z) (where z is an angle in radians measured from 
the positive axis of ordinates in the clockwise sense), 
defined as follows: /(;s)d2: is the length of time out of 
a complete scanning cycle during which the axis of 
vision is between the angles z and z dz.li the total 
time of one scanning cycle is T, obviously 

r f(z)dz = T; also f(z) ^ 0. (35) 

Jo 

It is now possible to obtain 7,, the instantaneous 
detection probability density resulting both from the 
instrument of detection and its use (method of scan). 
Let the angle f (from the positive axis of ordinates 
to the target. Figure 7 of Chapter 2) and the range r 
remain practically constant during one scanning 
cycle (slow relative motion or fast scan). Then at each 
cycle the length of time during which the target is 
within the angle e of the visual axis is 

I = 2€/(f) to terms of higher order in e. 

Jf- € 




44 


THE DISTRIBUTION OF SEARCHING EFFORT 


Hence the probability of detection is 


makes the expression 


g = 2e/(f)X(r). 

This is the one-glimpse probability of detection; in- 
deed, the idea of scanning automatically commits us 
to the notion of detection by discrete glimpses rather 
than by continuous looking. But in view of our as- 
sumption of a scanning which is fast with respect to 
the relative motions, it is a legitimate approximation 
to pass to the latter viewpoint, and to convert g 
into 7 , by division by T (compare with the similar 
reasoning in Section 4.4, equation (7) in particular). 
This we shall write 


7« = 7(r,f) = |/(r)XW. (36) 

It is to be noted that whereas in the greater part 
of Chapter 2, yt = t(Oj ^ function of range alone, 
the very nature of the present considerations focuses 
attention upon a 7 < which depends both on target 
range and bearing. Nevertheless, the relevant reason- 
ing and formulas of Section 2.4 are applicable, and 
we have as the expression for the search width 


W 




dx. (37) 


The mathematical nature of the problem is now 
clear. The quantities w, €, X(r) are fixed by the condi- 
tions, whereas T and /(^) are at our disposal, sub- 
ject only to the conditions (35) and that T must be 
small in comparison with the time required for an 
appreciable change in relative position of target and 
observer. And we have to find that function f{^) which 
makes W a maximum. 

By an easy argument of symmetry, the optimum 
/(f) is symmetrical about the axis of ordinates: 
/(-r) = /(r)- Assuming this, the integrand in (37) be- 
comes an even function of x, so that W is twice the 
integral from 0 to oo . Similarly, the integral in (35) 
can be replaced by twice its value from 0 to tt. Writ- 
ing for convenience 

m = I m, Kir) = ^X(r) 


(35) and (37) are further simplified, and our problem 
is reduced to the following: 

Find that function 0 ^ f ^ tt, which, among 
all functions satisfying 



m ^ 0, 


(38) 


W 

2 


f 1 - exp ^-f <l>{^)K{r)dyy^dx (39) 


a maximum. This is an “irregular” problem in the 
calculus of variations, not to be solved simply by 
equating certain variations to zero. 

Introducing polar coordinates (r,f), (39) becomes 




4>iOAi^ csc^)csc2^dr 


dx. 

(40) 


The integral with respect to x will increase when its 
(nonnegative) integrand increases; and such an in- 
crease will take place when the integral in the ex- 
ponent, i.e.. 


U 


£ 


(/>(^)A(x CSC t) csc^ 


is increased. The evident difficulty is that U involves 
X as well as If we fixed the value of x we could 
try to find the function 0(^) maximizing U subject 
to (38); but the resulting <^(^) could be expected to 
be one function for one value of x, and a different 
one for another x, and consequently useless as far as 
maximizing TF/2 is concerned. The remarkable fact 
is that in a broad class of cases, including all those 
important in practice, this turns out not to be the 
case. Indeed we have the theorem: 

U = Xi(r)/r2, where Xi(r) decreases with in- 
creasing r, the optimum scan consists in fixing the line 
of sight directly along the axis of abscissas, dividing 
the time equally between right and left. When most of 
the relative motion is due to the observer (e.g., a 
searching aircraft), this means that all scanning 
should be done abeam. 

To prove this theorem, we note that A(r) = 
A](r)/r^, where Ai(r) decreases as r increases. Hence 

V =Kf <#>((■) Ai (x CSC f) rff, 

^ Jo 


and our problem is to choose (/>(r) subject to (38) 
which maximizes this integral. As ^ goes from 0 to tt, 
X CSC f decreases from -|-oo to a minimum of x at 
f = 7r/2, and then increases to +«» again. Hence Ai 
has its maximum when f = 7r/2, and this is true for 
any a: > 0. The graph of Ai {x esc f) against^ is shown 
in Figure 2, which also shows that of 4>{^) [with shad- 
ing under it; the shaded area must be unity by virtue 
of (38)]. 




OPTIMUM DESTRUCTIVENESS 


45 


Geometrically, the problem is to deform the </)(r) 
curve (always maintaining it above the ^ axis and 
bounding unit area) so that the area under the prod- 
uct ordinate curve esc shall be a maxi- 

mum. Obviously, the more the 0(0 is peaked about 
the mid-point ^ = tt 1 2 the larger the area under the 



t 


Figure 2. Graphical representation of the scanning 
problem. 

product curve. In other words, the more of the time 
the axis of vision is directed at the angle f = 7r/2, 
the greater will be the value of the search width W. 
This proves the above theorem. 

The case X(r) = Xi{r)/r^, Xi(r) decreasing with in- 
creasing r occurs when \{r) is an inverse nth power 
with n greater than 2 , in particular, with the inverse 
cube power law; also, when there is an exponential 
attenuation factor multiplying an inverse square or 
higher power. In fact, it occurs in the majority of 
cases studied in Chapters 4, 5, and 6. And the theorem 
in the case of a definite range law is trivially true. 

It would be misleading to conclude that scanning 
should always be confined to the beam. In most cases 
it is imperative to detect the target early, for ex- 
ample, when it is a surfaced submarine which may 
submerge if not detected and attacked before it sees 
us, or in the case where it may be expected to attack 
us as soon as it sees us. 


3^ OPTIMUM DESTRUCTIVENESS 

There is a close analogy both in thought and in 
mathematics between the problem of the optimum 
distribution of searching effort and the optimum dis- 
tribution of gunfire, bombs, or other types of de- 
structive missiles which are projected into an area. 


To bring this out, it will suffice to consider a question 
which leads to results completely parallel to those of 
Section 3.2. 

Two areas Ai and A 2 contain targets of the nature 
of factories and other buildings important for the 
enemy’s economy. High-level night bombing is con- 
templated, and it is assumed that such bombing is 
sufficiently accurate to place the bombs in Ai or in 
Ao, but not to hit particular targets in these areas 
except by the operation of chance. Let the total 
vulnerable area of targets in Ai and A 2 be 5i and B 2 
respectively. Assume, furthermore, that the vul- 
nerability of all targets is the same, and that the re- 
sult of n random hits on the target area or R 2 ) 
is to reduce its effectiveness (value to the enemy) 
by a proportion 1 — A;” (where A; < 1). This means 
that if the Bi targets had a value to the enemy (pro- 
ductivity, rate of casualties it could inflict on us, 
etc.) expressible by a number E, the value after n 
hits on Ai would be reduced to k^E, the reduction 
in value being E — k^E, and hence the proportional 
reduction in value being the ratio of this quantity to 
E. Assume, Anally, that the value to the enemy of 
a set of targets is proportional to its total area. 

If N similar bombs are at our disposal and if it is 
as easy to drop bombs in A 1 as in A 2 , the decision as 
to how to divide the bombs between Ai and A 2 is 
properly made on the basis of maximizing the ex- 
pected damage to be inflicted. 

Let Ni bombs be dropped in Ai and N 2 in A 2 , 


N,-{-N2 = N, Ni ^0, A 2 ^ 0. (41) 

The probability that one bomb dropped in Ai will 
hit the target is BijAi. If Ni are dropped in Ai, the 
probability that Ri of them will hit the target is (by 
the binomial distribution) equal to 

Vi! 

Ri\ {Ni - Ri)\[a,) A,) 


The proportional damage done by the Ri hits being 
1 — k^\ the expected damage produced by Ni bombs 
dropped in A 1 is given by 


Ni 

Ri=0 


Nil 




the last equation being established by the algebra 


tfliiiiiiilfiiiiliinl MMM it Ti' 


46 


THE DISTRIBUTION OF SEARCHING EFFORT 


of the binomial theorem. It is convenient to write 
(for = 1 and 2) 


Mi 




(42) 


Bj (1 - h) 
Ai 


approximately, when Bi/Ai is small. 


With this expression, the expected (proportional) 
damage is 

1 - 

Since, as we are assuming, the value to the enemy 
is proportional to the target area, the expected loss 
of value to him when Ni and N 2 bombs are dropped 
in A I and A 2 respectively is proportional to 


V = Ai(l- + ^2 (1 - (43) 


This is the quantity which has to be maximized by a 
choice of Ni and N 2 satisfying conditions (41). Ob- 
viously the problem is mathematically identical with 
that of Section 3.2. Accordingly, it suffices to enumer- 
ate the results. 


Case 1 : yL 2 A 2 ^ piAi. 

All N bombs should be dropped in A 2 . 
Case 2\ jUiAi ^ M 2 A 2 . 


All N bombs should be dropped in Ai. 
Case 3: 

IM,A,e-'^^ < mAi 

and 

fXiAi e < fj, 2 A 2 . 


Then some bombs must be dropped in Ai and some 
in A 2 according to the formulas (for i = 1 and 2) 


N, = 


log Pi Ai 


+ 


-k- 

MiL 


log Ml A] 
Ml 


log M 2 A; 
M2 


- + - 
Ml M2 


(44) 


When N is very large, N 1 /N 2 = M 2 /mi> which ac- 
cording to (42) reduces to {B 2 /A 2 )/{Bi/Ai), so that 
the numbers dropped are inversely proportional to the 
probabilities of hitting the targets. 




Chapter 4 

VISUAL DETECTION 


41 INTRODUCTION 

T his chapter deals with the problem of visual 
search under the general conditions of illumina- 
tion which obtain during the daylight hours. The 
theoretical studies described are based on laboratory 
tests, operational data, and service trials. They at- 
tempt to answer two of the questions proposed in 
Chapter 2. These are, first: What is the maximum 
range within which a given target can be seen, and 
second: What is the chance that the target will be 
seen while it is at any given range? The answers re- 
quire some knowledge of the construction and per- 
formance of the eye considered as a detecting in- 
strument. In what follows in this chapter, therefore, 
the eye is studied first as a detecting instrument and 
then as applied to specific operational problems. 

42 the eye as a detecting instru- 
ment-general DESCRIPTION 

In general construction, the eye is very similar to 
a camera. The transparent front surface or cornea 
and the crystalline lens together constitute a com- 
pound lens which forms on the retina, at the back 
wall of the eye, an image of any given object in front 
of the eye. Between the cornea and the crystalline lens 
there is a small aperture known as the pupil. This 
aperture is variable in size over a limited range and 
determines the quantity of light which enters the eye. 

The retina corresponds to the sensitized plate or 
film in the camera. It contains two different types 
of sensitive elements or receptors known as rods and 
cones. The rods serve for night vision and are in- 
capable of distinguishing color. The cones are re- 
sponsible for vision in daylight and for all color 
vision. The central part of the retina, through which 
the visual axis passes, is known as the fovea. The 
visual axis makes a small angle with the optic axis 
of the compound lens system. The diameter of the 
fovea subtends an angle of between 1 degree and 2 
degrees at the effective center of the lens (actually 
the nodal point of the compound lens). The fovea 
is the region of most distinct daylight vision. It con- 
tains cones only, the average angular distance be- 


tween centers of adjacent cones being about 0.5 
minute of arc. As the angular distance from the axis 
increases beyond the edge of the fovea (i.e., as the 
parafoveal region is entered), the number of cones 
in unit area decreases, at first rapidly and then more 
slowly while the number of rods in unit area gradu- 
ally increases out to about 18 degrees and then de- 
creases. In daylight, therefore, a given target, can 
be seen most easily by looking straight at it while at 
night a better view is obtained by looking about 6 
degrees off from the most direct line of sight. 

Both rods and cones are capable of adjusting them- 
selves for the general level of illumination to which 
they are exposed. This adjustment is known as adap- 
tation. For both rods and cones adaptation is much 
more rapid to an increase in illumination than to a 
decrease. Furthermore, both light and dark adapta- 
tion are more rapid for cones than for rods. Dark 
adaptation for the cones takes several minutes, 
whereas for the rods this time is of the order of half 
an hour. 

Unlike the radar which scans continuously, the 
eye moves in jumps while searching and is capable 
of vision only during periods of little or no motion. 
These periods are known as fixations and last for 
about 0.25 second. In a given fixation or group of 
fixations, a target at extreme range can be seen in 
daylight only on the fovea so that the visual axis 
must be well within 1 degree of the line joining the 
target and the eye. As the range decreases, regions 
outside the fovea become capable of detecting the 
target, at first those near the fovea and then those 
farther out. Hence targets at less than extreme range 
can be seen not only on the fovea but off fovea as 
well, i.e., “out of the corner of the eye.’’ 

4 3 the eye as a detecting instru- 
ment-detailed PERFORMANCE 

The characteristics of the target and its back- 
ground, which determine whether or not the target 
can be seen, are : 

1. Brightness of the background. 

2. Brightness of the target. 

3. Color of the background. 


47 


48 


VISUAL DETECTION 


4. Color of the target. 

5. Size of the target. 

6. Range or distance of the target. 

7. Shape of the target. 

The background brightness, by reason of light 
adaptation, determines that differential sensitivity 
with which the eye discriminates as to differences in 
brightness between the object and its immediate sur- 
roundings. When the background is nonuniform, the 
sensitivity is set by an effective background. Possibly 
because of the fact that light adaptation is more 
rapid than dark, the contributions of the lighter 
portions of the field to the state of adaptation is 
larger than that of the darker portions. For a given 
target at a given range under daylight illumination, 
it is the magnitude of the difference between target 
and immediate background brightness, expressed in 
units of effective background brightness, which de- 
termines whether or not the target can be seen. This 
quantity is defined, for purposes of this chapter, as 
the brightness contrast. Under conditions of daylight 
illumination, i.e., for all illuminations greater than 
that of early twilight, a given contrast will cause the 
same visual response regardless of the magnitudes of 
the various brightnesses which go to make up this 
contrast. As the illumination is decreased below that 
of early twilight, the various brightnesses involved 
enter explicitly. Since this chapter is concerned with 
daylight illumination only, this further complica- 
tion is not considered. It is to be remembered that 
the definition of brightness contrast given above for 
purposes of this chapter is not the usual one found 
in the literature. 

It has long been believed that in comparison with 
brightness contrast, color is of little importance in 
determining whether or not a given target can be 
seen. Recent investigations' have supported this belief 
and have shown that any effects due to color can be 
ignored in most operational problems of visual search 
without thereby introducing any appreciable errors. 
With this as justification, all effects due to color are 
ignored in this chapter. 

The size of the target and its range combine to 
determine the solid angle which the target subtends 
at the eye and hence the size of the image on the 
retina. Hence the three characteristics of the target 
and its background upon which the discrimination of 
the eye depends under daylight illumination are: 

1. Contrast of the target against its background. 

2. Solid angle subtended by the target. 

3. Shape of the target. 


The two sets of measurements employed to de- 
termine the effects of these three variables are those 
of K. J. W. Craik“’^ and some unpublished measure- 
ments made in collaboration with Selig Hecht and 
Simon Shlaer at the Laboratory of Biophysics, Co- 
lumbia University.^ The Columbia experiments were 
designed primarily to determine the effect of object 
shape. While they are more detailed than those of 
Craik, they are not as extensive in retinal areas in- 
vestigated. The results of both sets of measurements 
constitute the primary laboratory data for this chap- 
ter. Craik’s measurements are employed to determine 
the effect of solid angle, while those made at Co- 
lumbia serve to determine the effect of target shape 
and as a check on the Craik experiments for those 
retinal regions for which both measurements are 
available. 

For any given target, the quantity measured in 
both the Craik and the Columbia experiments was 
the just-perceptible or threshold contrast. As is the 
case with any measurement, this quantity is not de- 
termined exactly but within some tolerance or un- 
certainty. In the Craik experiments, that contrast 
above which the target could always be seen and 
that below which it could never be seen were meas- 
ured and the average of these two was taken as the 
threshold contrast. In the Columbia experiments a 
frequency method was employed to determine the 
probability of sighting the target as a function of 
contrast. That contrast for which the sighting prob- 
ability was 57 per cent was taken as the threshold 
contrast. The results of the two sets of measurements 
are in excellent agreement so that the two experi- 
ments evidently measure the same quantity. 

From the results of the Craik experiments with 
circular targets it was found that the threshold con- 
trast Ct can be represented as a function of the solid 
angle co subtended by the target at the eye, by the 
following equation: 

Ct = a -\ — , (1) 

0 ) 

where a and b are constants for any one retinal region. 
Instead of using the solid angle co, it is often more 
convenient to employ the square of the visual angle 
a, i.e., the angle subtended at the eye by the diameter 
of the equivalent circle. The quantities a and b have 
different values at different angular distances from 
the center of the fovea. If 6 is this angular distance 
in degrees, from center of the equivalent circle to 
center of fovea; a, the visual angle in minutes; and 




CONTACT PROBABILITY AND SCANNING GENERALITIES 


49 


Ct the threshold contrast in per cent, Craik’s data 
can be represented by the following equation : 

C, = l,75r + — . (2) 

q,2 


The 'probability g is a function of the ratio C/Ct = 
{target's apparent contrast) / {target's threshold contrast 
alone), 


g 



( 3 ) 


The value of 6 which must be employed to obtain 
the foveal data from equation (2) is 0.8 degree and 
the threshold contrast is constant between 6 = 0 
degrees and 6 = O.S degree. It is to be pointed out 
that equation (2) is purely empirical. It represents 
the Craik experiments with fair accuracy. Since the 
number of measurements made by Craik is relatively 
small, the experimental error is fairly high: Hence 
slight modifications of equation (2) are to be expected 
when the more extensive Columbia experiments are 
complete. The reader will note that in the present 
chapter a and 6 are used in an altogether different 
meaning from elsewhere in this book. 

The targets employed in the Columbia experi- 
ments were all rectangles and the quotient q, the 
ratio of length to width, was taken as a measure of 
the asymmetry. This quotient q is known as the 
asymmetry factor. The results of these experiments for 
the different background brightnesses and for the 
various retinal regions investigated all show the same 
general trends. As the shape of a small target is 
changed, keeping its angular area constant, the 
threshold contrast remains constant until the asym- 
metry factor reaches a value such that the angular 
length of the target is about 3 minutes of arc. As the 
asymmetry factor is increased beyond this point, the 
threshold contrast gradually increases. As the shape 
of a large target is changed, keeping its angular area 
constant, the threshold contrast gradually decreases 
until the asymmetry factor is such that the angular 
width of the target is about 2 minutes of arc. Beyond 
this point, the threshold contrast again increases. 
The greatest effect of asymmetry is observed when 
the angular size of the target is about 10 square 
minutes. As the asymmetry factor is changed from 
2 to 200, the threshold contrast increases by a factor 
of 4. Between 2 and 100 the factor is 3. 

Two further fundamental facts have been es- 
tablished by the Columbia experiments : If the posi- 
tion of the target is known so that no search is re- 
quired, the probability of sighting the target is 
independent of the time of exposure provided this 
time exceeds that required for a single fixation; and 
the ‘‘glimpse” probability g that a target be sighted 
at a single fixation obeys the following law : 


and only through Ct do the target's apparent size a and 
the off axis angle 6 intervene. The quantity g is pre- 
sented in Figure 1 as a function of C/Ct. This is 
the experimental curve for the function /. 



Figure 1. “Glimpse” probability of detection as a func- 
tion of apparent contrast /threshold contrast — experi- 
mental data. 


44 CONTACT PBOBABILITY AND SCAN- 
NING— GENEBALITIES 


Let a given target be viewed under given condi- 
tions: 6, and its apparent size (visual angle) and 
contrast, a and C, are therefore given. The latter 
determine a threshold value 6q of the angle from the 
line of sight: mathematically, by the law expressed 
in (2), i.e., Oq is defined by the equation 

C = 1.759o‘ + — . 

On the other hand, 6 and a determine a threshold 
contrast Ct by the same law (2) : 


Ct 


l.lbS^ + 


m 


On combining these equations with (3), the following 
is derived. 


g=.f 


1.759„i+l^ 

a 


1 . 750 ^ + 


190 


( 4 ) 


The philosophy of this method of obtaining the ex- 


50 


VISUAL DETECTION 


pression (4) for g may be described as follows: The 
triplet of values C, 6, and a determines the required 
probability g. To find the unknown g, we contemplate 
successively the triplets Ct, 0, and a and C, do, and 
a, each determining the standard threshold prob- 
ability 0.57 (by determining do from C and a, and 
Ct from d and a, by (2) to give this threshold prob- 
ability). Because of the fundamental equation (3), 
this effectively furnishes the required g in terms of 
C, d, a. 

Now consider what happens when the conditions 
of the last paragraph are varied as follows: The ap- 
parent size and contrast, a and C, remaining fixed, 
the angle d is varied. Since do is a function of a and 
C, it will stay fixed. Thus the letters do and a in 
(4) represent unchanging quantities, d alone vary- 
ing. Consequently (4) expresses the manner of de- 
pendence of the one-fixation chance of sighting g upon 
the angle d off the line of sight. 

For targets of very small apparent size {a < < 10), 
(4) reduces to g = f{do/d), a graph of which is shown 




0 12 3 4 


Figure 2. “Glimpse” probability of detection as a func- 
tion of off-axis angle /threshold off-axis angle, for tar- 
gets of (A) very small, and (B) very large apparent size. 
9o/d is the X axis. 


quantities C and a are assumed to be independent 
of the position on L. The eye fixates once upon a 
random point on L. If the angle subtended at the 
eye by L is © degrees, it is easily shown that when 
© is large g is given with sufficient approximation 
by the formula 

2 

g = 0 j^ gde. (5) 


For the chance that the point of fixation 0 be close 
to the ends of L can be neglected, so that it can be 
assumed that L extends a considerable distance on 
either side of 0. The probability that the target be 
between d and d dd degrees away from 0 is d^/©. 
Hence the chance of sighting is the integral of gdd / © 
over the whole angular range ©. But since g falls to 
zero at appreciable angles from 0 [as can easily be 
shown on the basis of equation (4) and because of 
what has been said concerning the distance of the 
extremities of L from 0], the limits of integration 
can be taken as — <» and ^ [g being defined as 
zero for values of d beyond those contemplated in 
(4)]. By symmetry of the integrand, twice the integral 
between the limits 0 and + °° can be used; hence 
the validity of equation (5). 

In combination with (4), (5) becomes 



1 . 7500 ^ + 


1900 


1.750^ + 


190 


dd 



1 . 7500 * + 


1.750o*X* + 



d\ 


in Figure 2 A. For targets of very large apparent 
size (a > > 10), (4) becomes g = f (•%/ 0o/0) , plotted 
in Figure 2B. 

Having considered the dependence of g upon 0, 
it is now necessary to evaluate the one-fixation prob- 
ability g when the value of 0 is unknown hut distributed 
at random with a known frequency, C and a still being 
given constants. The circumstances which impose 
this necessity are the use of the eye to scan a given 
location within which the target is supposed to be, 
but in an unknown position. Two cases are important 
in practice: 

1. Linear Scan. The target is located on a line L 
and is uniformly distributed thereon; the pertaining 


= ( 00 , 0 :), 

where the new variable of integration X = 0/ 0o. The 
coefficient A (0o,o:) has in the extreme cases a < < 10 
and a >> 10, the values 

respectively, and intermediate values of a are found 
by graphical integration to lead to intermediate 
values of A (0o,o:). To the degree of approximation 
which is permissible, it can therefore be assumed to 


CONTACT PROBABILITY AND SCANNING — GENERALITIES 


51 


be independent of Bq and a. We shall assume hence- 
forth that A = A (Bo, a) = 2.36. This leads to the basic 
proposition : 

The prohahility g is proportional to the threshold 
angular distance Bq of the target from the visual axis, 
and is inversely proportional to the angle © subtended 
by the linear locus L of target positions: 

^ 2.36^0 (g) 

^ 0 

It is now expedient to modify the point of view, 
and instead of regarding the act of sighting as pro- 
ceeding by a rapid succession of fixations (glimpses), 
to envisage it as progressing continuously in time 
(continuous looking). If dt is an interval of time 
which is short compared with the time taken for the 
observer and target to change relative positions by 
an appreciable amount, and short also in the sense 
that the chance of a detection during it is small, but 
large in comparison to the time of a single fixation 
(actually, one quarter of a second), it becomes 
legitimate to consider ydt, the probability of detec- 
tion between the epochs t and t -j- dt. Since when 
dt = second, ydt = g, we have y = 4 i.e.. 


7 = 


9.4400 

© 


(time measured in seconds) 


= 9.44 X 


60^00 

© 


(time measured in hours). 


( 7 ) 


According to equation (4) of Chapter 2, the mean 
^‘pickup time’’ (time required to detect) is I/7. This 
provides a method of measurement of 7. Craik^ per- 
formed a series of laboratory experiments measuring 
mean pickup time with line scanning, with © = 45 
degrees. The results are shown in Figure 3, where 7 
is plotted against Bq. The approximate linearity is 
evident, in accordance with (7). But the coefficient 
in (7) has to be multiplied by about 0.13, a dis- 
crepancy which could be accounted for on the as- 
sumption that the fixations in Craik’s experiments, 
instead of being arranged at random, occur effectively 
in groups of 7 or 8. We will return to this coefficient 
in a later place in connection with the practical ap- 
plications. 

2. Area Scan. The target is located in a region of 
a plane, subtending the solid angle 12 square degrees. 
The distribution is uniform over the area, and the 
quantities C and a are independent of position. 
Reasoning precisely similar to that employed earlier 
(with double integration replacing single integration) 


establishes the formula for the one-fixation prob- 
ability : 




( 8 ) 


Introducing (4) with the variable of integration 
X = d/6o leads to 


-g = ^ M{e„,a) 


M{9o, 


a) = 2^ f f 
Jo 


1.759„* + 


199o 


1.75eo*V + 


199oX 


( 9 ) 

XdX. 



Figure 3. Instantaneous probability of detection as a 
function of threshold off-axis angle — experimental data 
by Craik.3 


The extreme values of this quantity occur when 
a < < 10 and a > > 10; they are respectively 

2>rJ^/(i)xdX = 4.16 and 2rff(^^^\d\ = 6.38. 

Correspondingly, y is given by 
0 ^ 

7 = ^ 4M (00,0:) (time in seconds). (10) 
We shall return to these expressions later. 


52 


VISUAL DETECTION 


45 TARGET AND BACKGROUND 
—GENERAL 

In the preceding sections, the eye has been con- 
sidered as a detecting instrument operating on tar- 
gets of given apparent size and contrast (and shape, 
in so far as this is relevant), and the laws governing 
the probabilities of detection have been set forth. 
Before these results can be applied to an actual case 
(viz., a naval operation), it is necessary to find how 
the circumstances of the case determine these vari- 
ables and hence, indirectly, the probabilities of de- 
tection. Now in any actual case, certain intrinsic 
characteristics of the target may be regarded as 
known. These are its geometrical shape and dimen- 
sions and its diffuse reflecting power. Except within 
small angular distances from the sun, the latter de- 
termines the intrinsic brightness of the target in 
units of sky brightness. The intrinsic characteristics 
of the immediate and general background and the 
relationships between all intrinsic and apparent 
quantities are determined by the circumstances of 
the case. The study of these various dependences and 
relations is the subject of the two following sections. 

45 DEPENDENCE OF APPARENT CON- 
TRAST ON ATMOSPHERIC CONDITIONS 

The presence of haze in the atmosphere alters the 
pattern of light received by the eye from the various 
points of the target and background. It acts in two 
ways: 

1. The haze removes some of the light by absorp- 
tion and scattering out of the line of sight. 

2. The haze adds some light reaching the eye from 
the direction in which the observer is looking by 
scattering into the line of sight. 

In order to work out the general equations, let 
B = the apparent brightness of the target. 

Bo = the intrinsic brightness of the target, 

B^ = the apparent brightness of the background, 
Bf, = the intrinsic brightness of the background, 

Bs = the brightness of the sky, 

= the atmospheric scattering coefficient, 

V = the meteorological visibility, 

R = the target range. 

The apparent brightness of the target is 

B = Boe-^^ + BAI - 

where the first term represents the light reaching the 


eye directly from the target, and the second the light 
scattered into the eye by the haze between the target 
and the eye. Similarly the apparent brightness of the 
background immediately surrounding the target is 
given by 

= B^e-^^ + R, (1 - 6"^^). 

The difference in brightness between object and im- 
mediate background is, therefore, 

B - B' = (Bo- Bo) e-'*". 

The apparent contrast is the ratio of this brightness 
difference and the effective background brightness in 
accordance with the discussion of brightness contrast 
given in Section 4.3. 

There are two main cases which arise in actual 
operations, one in which the immediate background 
and the effective background are the same and one 
in which they are different. The first case is exempli- 
fied by search, from land or from a surface ship, for 
a surface ship silhouetted against the sky. The second 
case is exemplified by search from the air for a target 
on the sea. In this case, the line of sight frequently 
approaches the horizon so that the adaptation of 
the eye is determined partly by the sea brightness 
and partly by that of the sky. Because of the fact 
that light adaptation is rapid compared to dark 
adaptation, the bright sky is responsible, almost en- 
tirely, for setting the level of response of the eye. 
Hence in both cases, the sky brightness is a reason- 
able approximation to the effective background 
brightness and hence 

c = (11) 



is the intrinsic contrast as defined earlier for pur- 
poses of this chapter. It is to be remembered that 
within 10 or 15 degrees of the sun, the level of re- 
sponse of the eye is altered by the sun’s glare so that 
the effective background brightness is somewhat 
greater than the average sky brightness Bg. This 
effect is neglected in the present theory but must be 
considered in any exact treatment. 

The quantity usually quoted as a measure of at- 
mospheric conditions is not the scattering coefficient 
jS but the meteorological visibility V. It is desirable, 
therefore, to express equation (11) in terms of the 
meteorological visibility rather than in terms of the 
scattering coefficient. The meteorological visibility is 


MAXIMUM SIGHTING RANGE 


53 


usually defined loosely as the maximum distance at 
which large targets such as mountains or high coast 
lines can be seen against the sky. Merton® gives 78.3 
per cent as the intrinsic contrast of such targets and 
2.5 per cent as the threshold contrast. Substituting 
C = 2.5; Co = 78.3 and R = V in equation (11), 
^ = 3A4:IV. From this it follows that 

C = (12) 

Although the threshold contrast measured in the 
laboratory is often less than 2.5 per cent, an examina- 
tion of operational estimates of the meteorological 
visibility indicates that 2.5 per cent is a good ap- 
proximation to the practical operation figure. 

47 INTRINSIC BRIGHTNESS OF THE SEA 

The light flux reaching the eye from the sea con- 
sists of two parts, that reflected specularly and that 
reflected diffusely. If the sea is perfectly calm, these 
two combine to produce an intrinsic sea brightness, 
4 per cent that of the sky immediately below tiie ob- 
server and increasing gradually to 100 per cent that 
of the sky at the horizon. Measurements of the 
sky /sea brightness ratio® indicate that even in mild 
seas the mirror surface characteristic of a perfectly 
calm sea is so broken as to cause wide departures 
from calm sea conditions except within relatively few 
degrees of a low-lying sun or moon. For sea states 
other than calm, the intrinsic sky/sea brightness 
ratio is fairly constant from the horizon to about 
45 degrees below and has a value of about 2. For an- 
gles greater than 45 degrees below the horizon, the in- 
trinsic brightness of the sea gradually decreases from 
about 50 per cent at 45 degrees to near 4 per cent 
directly below the observer. 

48 MAXIMUM SIGHTING RANGE 

With the information in earlier sections concerning 
the characteristics of target and background on which 
visual detection depends and the influence of the 
operational situation on these characteristics, it is 
possible to obtain answers to the two questions pro- 
posed in the introduction. The first of these is: What 
is the maximum range within which a given target 
can be seen? The quantity which is here defined as the 
maximum range is that range at which the target con- 


trast reaches the foveal threshold. As was the case with 
the threshold contrast discussed in Section 4.3, the 
maximum range is not determined exactly but within 
some tolerance or uncertainty, so that some targets 
viewed foveally will be seen beyond this range while 
others within this range will be missed altogether. 
We will see, later in this section, the magnitude of 
this uncertainty, i.e., the spread of ranges over which 
detection foveally is not a certainty. It is the pur- 
pose of this section to determine the maximum sight- 
ing range in terms of those quantities which occur 
in the operational situation. We have given a target 
of actual area Ao, apparent area A ; intrinsic contrast 
Co and asymmetry factor q viewed through an at- 
mosphere in which the meteorological visibility is V. 
In terms of the apparent area A and the range R, 
the visual angle a, defined in Section 4.3 is given by 

« = 0.64'^, (13) 

R 

where A is in square feet, R in nautical miles and a 
in minutes of arc. 

Consider first the case in which the target is cir- 
cular and the meteorological visibility is unlimited. 
Let Ro be the maximum sighting range under condi- 
tions of unlimited visibility. Substituting equation 
(13) in equation (2) with do = 0.8, its value for foveal 
vision, as discussed in Section 4.3, is 

R« = 0.164 [(Co - 1.57)A]*. (14) 

In terms of the actual area of the target Aq, there 
are two cases which occur frequently in operational 
situations, one in which the target is viewed at ap- 
proximately normal incidence, and the other in which 
the target, flat on the surface of the sea, is viewed 
at an angle. For the first case, the apparent and 
actual areas of the target are equal, so that 

Bo = 0.164 [(Co - 1.57)4o]* . (15) 

If h is the altitude of the observer in feet, then for 
the second case A = 1.64 (10“^)Ao/i/i?o. Substituting 
in equation (14) we have for the second case 

Ro = 1.64 (10“') [(Co - 1.57)Aoh]\ (16) 

Consider next a circular target viewed through an 
atmosphere having a meteorological visibility V and 


54 


VISUAL DETECTION 


let Rm be the maximum sighting range. The con- 
trast is given by equation (12). Making the indicated 
substitutions in equation (2) leads to transcendental 
equations for case I and case II. These can be solved 
for the meteorological visibility V in terms of the 
maximum sighting range Rm- These equations are, for 
case I, 


V = 


1.49/?. 


, r CoAo 1 

[ 30.9 Rj + 1.57 AoJ’ 


(17) 


and for case II, 
V = — 


1.49 /?. 


logi 


f 1, 

[2.26 (10'>)fi„« + 1.57 AoftJ 


which can be rearranged to give 

1.49(fi„A^) 

r ■ 

°®“[ 2 - 26 ( 10 ®) (RJ/h) + 1.57 


(18) 


The effect of target asymmetry reaches its maxi- 
mum in the following operational situation : A target 
of the case II type having a ratio of apparent length 
to apparent width of 100, observed at a range such 
that the solid angle subtended by the target is 4.5 
square minutes under conditions of unlimited me- 
teorological visibility. Under these conditions the 
maximum range turns out to be half what it would 
be for a perfectly symmetrical target (disk) of the 
same intrinsic contrast and solid angle subtended. 
Except in such rare cases as the extreme just de- 
scribed, the effect of target asymmetry on maximum 
range is small. This is true not only for the maxi- 
mum range, but for all similar quantities (e.g., do). 
In the interest of simplicity, therefore, and with only 
slight loss of accuracy, the effect of target asymmetry 
shall he neglected henceforth. 

We are now in a position to investigate the un- 
certainty in the range at which a target, viewed 
foveally, can be seen. This is done by computing the 
ratio of apparent to threshold contrasts for each of 
a number of ranges and looking up the sighting prob- 
abilities corresponding to these ratios from Figure 1. 
[Essentially, the use of equation (3).] Sighting prob- 
abilities, determined in this way, are presented in 
Figure 4 as functions of the ratio of range to maximum 
range as defined earlier in this section. The curves A, C 
are for small high-contrast targets seen under condi- 


tions of unlimited meteorological visibility while the 
curves B, D are for large targets seen through dense 
haze. From these curves it would appear that, as 
measurable quantities involving human behavior go, 
the maximum sighting range is a quantity which 



T 


\ 



\ 

B 

\ 


0.8 1.0 1.2 1.4 0.8 1.0 1.2 



0.8 1.0 1.2 1.4 0.8 1.0 1.2 


R/Rm 


CASE n 


R/Rm 


Figure 4. ‘‘Glimpse” probability of detection as a func- 
tion of target range /maximum range. 


can be determined to within a relatively small un- 
certainty. 

In order to compute the maximum sighting range 
under some specified set of conditions, using equa- 
tion (17) or equation (18) as the case may be, it is 
necessary to know the area and intrinsic contrast of 
the target. These constants have not yet been deter- 
mined for any large number of targets. However, 
there is one large class of targets for which these 
constants are known reasonably well, i.e., surfaced 
ships as viewed from an aircraft. Here the target 
seen first in the vast majority of the cases is the wake. 
As will be seen in a later section, the intrinsic con- 
trast of a wake is about 50 per cent. Three general 
classes of ships are considered here, surfaced sub- 
marines, ships having medium-sized wakes, and those 


VISUAL PERCEPTION ANGLE PROBABILITY OF DETECTION 


55 


having large wakes. The second class includes de- 
stroyers, destroyer escorts, and medium-sized mer- 
chant ships, while the third class includes large com- 
batants and high-speed ocean liners. The wake areas 
for the three classes are given approximately as 
l.SxlO'^, 1x10®, and 2x10® square feet, respectively. 
In Figure 5, Rm/h^ is presented for each of these cases 



Figure 5. Variation of maximum range with visibility 
for three types of target (naked eye) . 


as a function of V /h^ for naked eye search. In Figure 
6, the same quantity is presented for search with 
U. S. Navy standard 7x50 binoculars. The only effect 



Figure 6. Variation of maximum range with visibility 
(7x50 binoculars). 


of the binoculars which has been considered is that 
of its magnifying power on the apparent target area. 
Because of the neglect of the slight reduction in 


apparent contrast (beyond that due to haze) which 
results from the fact that the binoculars are not per- 
fectly transparent, the results presented in Figure 6 
are slightly optimistic as regards maximum sighting 
range. 


VISUAL PERCEPTION ANGLE— PROB- 
ABILITY OF DETECTION 

As observed in Section 4.4, the fundamental ele- 
ment 7 , in any study of the probability of sighting can 
be expressed in terms of Bq, which can he considered 
as the visual perception angle. The equations relating 
7 and Bq are equation (7) and equation (10) for linear 
scan and for area scan respectively. Just as was the 
case with the maximum sighting range discussed in 
the last section, so the visual perception angle Bo can 
be obtained from equation (2) in terms of the vari- 
ables which occur in the operational situation, by 
making the proper substitutions for visual angle a 
and contrast C. This leads, in the case of a target 
viewed normally, to an equation for Bq involving A, Co, 
V, and R, and of a target on the sea surface, to an 
equation involving A, Co, V, h, and R. If, instead of 
the variables listed above, the variables R/Rm, R^nlV, 
and Co are employed, the number of variables is re- 
duced, in case I from 4 to 3, and in case II from 5 to 3. 
These changes of variables Avill now be made. Sub- 
stituting in equation (2) for C from equation (12) and 
for a from equation (13) leads to, 

for Case I, 

l.759„» + (19) 

Ao 

If 7^ = R,n, equation (19) becomes 

g-3.44 R./V ^ J gg5 ^ ^712 Rj (20) 
Ao 


Eliminating Ao between equations (19) and (20) gives 


Coe' 


■SAMR/Rm) (Rm/V) 


- 1.750 


CoC 


3.44 Rm/V 


1.565 


= 1.25(AY,„. 

\rJ 


(21) 


For Case II, a similar procedure leads to 


Coe 


-SAMR/Rm) (Rm/V) 


1.750c 


Cog-3.44(/2WF) _ ^ 


= 1.25(A)\. 

\rJ 


( 22 ) 


56 


VISUAL DETECTION 


From these equations it follows that 


where 



and 


G = 


0.8 {Rm/RY 

(^Og -3.44 S„/F _ I 5g5 


(23) 


The constant n has the value 2 for Case I and 3 for 
Case II. 



A R/Rm 

Figure 7A. Threshold off-axis angle as a function of 
range /maximum range. (Case II: Co = 50 per cent.) 


As an example, values of Bq for wake-type targets 
are presented in Figure 7A as functions of RIRm- It 
is to be remembered that a wake is a Case II target 


having an intrinsic contrast of 50 per cent. The three 
curves shown are for three different values of Rm/Vi 
0, 0.5, and 1. The value RmlV — 0 corresponds to 
unlimited meteorological visibility (no haze) while 
Rm/y — 1 corresponds to haze so dense that the 
maximum range is determined by the haze alone. 



Figure 7B. Contours of 57 per cent detection proba- 
bility. 

The curves shown in Figure 7 A have all been termi- 
nated at 90 degrees. This is about the maximum 
value of ^0 for which vision is possible. 

It is interesting to note that equation (23), as a 
polar equation expressing the functional relationship 
between Bq and R/Rm, describes a surface of revolu- 
tion which is generated by rotation about the line 
of sight. Within this surface the contrast of the given 
target is above threshold and along the surface the 
probability of detection in a single fixation is con- 



SUBMARINE AND SURFACE SHIP SIGHTINGS FROM AIRCRAFT 


57 


slant and equal to 57 per cent. This constant prob- 
ability surface of revolution can be thought of as a 
detection lobe analogous to the equal power surface 
which describes the radar detection lobe. Visual 
scanning, therefore, can be thought of as an at- 
tempt to so align the detection lobe as to cause the 
target to fall within it so that detection can take 
place. As examples, central sections through lobes 
corresponding to the three curves of Figure 7A are 
presented in Figure 7B. 

If the type of scan employed is known so that it can 
be classified as line scan or area scan, then the in- 
stantaneous probability of detection 7 can be ob- 
tained from the visual perception angle do by means 
of equation (7) or equation (10), as the case may be, 
with one reservation: The constant in either equa- 
tion must be determined from the circumstances of 
the case, i.e., the number of lookouts employed and 
the fraction of the time during which each is search- 
ing effectively. In some cases it is possible to deter- 
mine, from operational or test data, an overall con- 
stant for the particular search situation. 

Thus the second fundamental question finds itself 
answered: The 'probability that the target will be seen 
has been obtained. 

4 10 SUBMARINES AND SURFACE SHIPS 
AS SIGHTED FROM AIRCRAFT 

With the material presented in earlier sections of 
this chapter and that in other pertinent chapters of 
this book, we are now prepared to analyze some 
actual operational situations. In this section, the 
sightings of submarines and surface ships from air- 
craft are examined to illustrate the application of the 
methods so far developed and to check some of the 
conclusions of earlier sections of this chapter. The 
emphasis, in this section, is on submarine sightings, 
since this is the situation for which the most com- 
plete operational information is available. 

The sightings of submarines and surface ships from 
aircraft come under the Case II classification of Sec- 
tions 4.8 and 4.9, since it is the wake which is sighted 
first in the vast majority of cases. In order to de- 
termine the maximum sighting range Rm and the 
visual perception angle do, it is necessary to know 
two constants characteristic of the particular target 
and one characteristic of the surroundings alone. 
These are the intrinsic contrast Co, the area Ao, and 
the meteorological visibility V. An estimate of the 


last is usually given in any operational sighting re- 
port. 

The wake of a ship is an excellent diffuse reflector, 
sending back practically all the light which falls on 
it so that its brightness approaches that of the sky 
from which it is illuminated. From Section 4.7, it is 
clear that the sea brightness is half that of the sky 
unless the sea state is glassy or the line of sight ‘is 
within about 10 degrees of the sun. From Section 
4.6 it is seen that the effective background bright- 
ness is that of the sky, unless the line of sight comes 
within about 10 degrees of the sun. Hence from the 
definition of intrinsic contrast given in Section 4.3, 
Co is about 50 per cent except under the rare condi- 
tions of sea and sun mentioned above. These rare 
conditions are neglected here. 

There are two methods of obtaining the target 
area Ao both of which are capable of reasonably high 
accuracy when applied to data taken under the ideal 
conditions which usually obtain during service trials, 
and usable when applied to operational data. In the 
first method, the wake area is obtained from measure- 
ments of photographs taken from the air. Several 
submarine photographs, taken during operational 
sorties, were measured and the values ranged from 
7,000 to 15,000 square feet, with an average of 11,000 
square feet. 

In the second method the area is obtained by 
means of equation (18) from measurements of the 
maximum range and the meteorological visibility. In 
service trials these two measurements can be made 
with fair precision. Under operational conditions, 
the ranges are distributed between zero and maxi- 
mum and the meteorological visibility is only esti- 
mated, so that the required quantities must be ob- 
tained indirectly. The uncertainty concerning the 
meteorological visibility is overcome by considering 
only those sighting reports in which the effect of 
atmospheric haze can be neglected, i.e., those in 
which the meteorological visibility was estimated to 
be unlimited. Under these conditions equation (18) 
reduces to equation (16). To obtain Ro a histogram 
was plotted with R/h^ as abscissa and the number in 
a specified interval of the abscissa as ordinate. The 
maximum value which R/h^ can have for ordinates 
greater than 0 is Ro/h^. The value of Ro/h^ obtained 
from the histogram of submarine sightings is 1.4, 
which, when substituted in equation (16), gives Ao 
= 1.3(10^) square feet, the value quoted in Section 
4.8. This value is believed to be more accurate than 
the 1.1(10'*) square feet obtained by the other 


58 


VISUAL DETECTION 


method. The photographic method was applied to 
ships’ wakes to obtain the values of 10^ and 2(10^) 
square feet for medium and large ships respectively, 
as quoted in Section 4.8. No great accuracy is claimed 
for these last two values of target area. 

In order to obtain 7, the instantaneous probability 
of detection, it is necessary to know first the type 
of scan employed and hence the functional relation- 
ship between 7 and the visual perception angle do, 
and second the constant of proportionality which re- 
lates 7 to the required function of In order to de- 
termine the best method of scan, a little more in- 
formation is required concerning the operational situ- 
ation. An examination of the submarine sighting data 
shows that the altitude of flight is always small com- 
pared to the maximum range. If both are expressed 
in the same units, the average ratio is 0.06. Under 
these conditions, if fixation is at a point on the ocean 
distant Rm from the aircraft, then any target be- 
tween this point and that directly under the aircraft 
is well within the visual perception angle do. Indeed 
this is still the case if the altitude is 0.10 times the 
maximum range or 600 feet per nautical mile of 
maximum range. 

The greatest chance of sighting a given target is 
obtained if the aircraft is made to do the scanning. 
This scanning is done by simply keeping the fixa- 
tions, on each side of the aircraft, within a small 
angular distance from a point on the surface of the 
ocean, directly abeam and distant from the air- 
craft. With this scheme, every target must pass 
through the visual perception angle do and if it stays 
within this angle for more than one-quarter of a 
second, detection by the eye is practically certain, 
given, of course, complete attention: missing of any 
targets under these conditions is due to the observer’s 
lack of attention. 

The system just outlined has been found the best 
possible for hunting friendly targets such as life 
rafts. For enemy targets, however, it is desirable to 
sacrifice some chance of sighting the target in order 
to increase the range at which it is most likely to be 
seen. A first approximation to the best compromise 
for enemy targets is a uniform scan along a line on 
the ocean surface distant R^ from the aircraft. It is 
recommended that scanning be confined to the front 
180 degrees of azimuth only. The chance of sighting 
the target under these conditions is the same for 
surfaced targets as if the scanning were uniformly 
distributed over the entire 360 degrees, for, whereas 
the scanning azimuth is halved, the effort over that 


azimuth is doubled. Employing the first 180 degrees 
only, therefore, results in no loss of targets, and has 
the advantage of early target detection. 

At the altitudes usually employed, less than 600 
feet per nautical mile of maximum range, the angular 
departure of any target inside the maximum range 
circle from the scanning line is so small compared to 
the visual perception angle do that no great error is 
made by considering all targets as being situated on 
the scanning line. Hence the problem under con- 
sideration is one of pure line scan with 7 proportional 
to do. 

To obtain the constant of proportionality between 
7 and do it is necessary to resort to the operational 
data. The quantity which can be obtained from these 
data is the sweep width W, defined in Chapter 2 as 



where px is the probability of detecting a target 
distant x from the aircraft track. To do this the 529 
submarine sightings on which reports were available 
were divided into groups representing ranges of R^- 
For each group a lateral range distribution curve was 
plotted and normalized to unity in the center, (i.e., 
at a: = 0) . This normalization to unity in the center 
is equivalent to saying that it is virtually impossible 
to fly directly over a target as large as a surfaced 
submarine without seeing it — in spite of such rare 
occurrences as that mentioned in Section 2.2. There 
is good evidence in support of this view.^ The area 
under the lateral range distribution curve so normal- 
ized is pxdx which determines W. The values of 
W obtained from the operational data are presented 



Figure 8. Path swept /maximum range as a function 
of maximum range — comparison of computed (solid 
line) and operational (dotted line) results (surfaced sub- 
marines) . 

in Figure 8 as a function of Rm- These values are in 
units of Rm. and the dots are the operational points. 

In order to compute W, it is to be recalled that, 
from Chapter 2, 


SUBMARINE AND SURFACE SHIP SIGHTINGS FROM AIRCRAFT 


59 


Vx = 1 




(25) 


and equation (7) y = kdo. The constant k for the 
operational situation under consideration is as yet 
unknown since it depends upon the number of look- 
outs, the fraction of lookout time spent in search, 
and the degree to which the search approaches uni- 
formity over the scanning line considered. If y is 
the distance, in nautical miles, of the target from the 
point of nearest approach to the aircraft and v is the 
aircraft velocity in knots, dt = 3,600 dy/v in seconds, 
so that 




edt 



Expressing dy in units of Rm, 



3,600 Rr 


r ■ 

Jo 


(26) 


(27) 


The integration indicated in equation (27) has been 
carried out graphically for various values of x/R^^. 
These are presented in Figure 9 as functions of xjRm. 
The two curves shown are for RmlV = 0.5 and Rm/V 
= 1, respectively. The values of do employed for the 
integrations were obtained from Figure 7A. In the 
operational data Rm/V = 1 for values of Rm = 5 
nautical miles and 0.5 for Rm = 10 nautical miles. 
The aircraft velocity for the 529 incidents averaged 
135 knots. 

To obtain k, the following procedure was em- 
ployed: For Rm = 5, Rm/V = 1, and a given value 
of k, ydt was computed from equation (27) and 
the integral given in Figure 9 for each of a number 
of values of x/Rm- For each of these, px was com- 
puted from equation (25) and a lateral range curve 
was plotted. From the lateral range curve a value of 
W/Rm was obtained by graphical integration. This 
procedure was repeated for a number of values of k 
until one was found which fitted the operational data 
of Figure 8. The value of k obtained in this way is 
1.75 X 10-3. 

Various lateral range curves, computed using k = 
1.75X10-3 and v = 135 knots, are presented in 
Figure 10. The quantity W/Rm obtained from these 
curves is presented in Figure 8 as a function of Rm- 
An examination of Figure 8 shows good agreement 
between the computed and the operational points at 
low values of Rm and a gradual separation of the 



0 0.2 0.4 0.6 0.8 1.0 1.2 


V^m 

Figure 9. Integral of equation (27) as a function of 
lateral range /maximum range. 



Figure 10. Lateral range distribution as a function of 
lateral range/maximum range, assuming A: = 1.75 X 10“®, 
y = 135 knots (surfaced submarine). 


60 


VISUAL DETECTION 


two as Rm increases. Now the theoretical curve is for 
ships which remain surfaced, while the experimental 
curve is for submarines which can evade detection 
by diving. This effect has been investigated^ and it 
has been shown that the effect of this evasion on the 
path swept W is negligible for values of ^ 5 



Figure 1 1 . Path swept as a function of maximum range, 
assuming y = 150 knots. 


nautical miles and gradually increases to about 35 
per cent for = 10. This tendency is apparent in 
Figure 8. 

In Figure 11, W for 50 per cent intrinsic contrast 
targets is presented as a function of Rm for naked- 
eye search. The more modern figure of 150 knots has 
been taken as the aircraft velocity. To obtain W in 
any particular case, Rm is first obtained from Figure 
5 and then W for the given Rm is obtained from 
Figure 11. It is to be remembered that altitudes greater 
than 600 feet per nautical mile of maximum range are 
not recommended. 

The greater values of Rm found in Figure 6 for 
7x50 binocular search might, at first sight, lead to 
the belief that W should be greater for binocular than 
for naked-eye search. It is to be remembered, how- 
ever, that inside a binocular field of magnifying 
power 7, the eye must search a scanning line 7 times 
as long. Hence for the same search effort, k is 1/7 
as great. The quantity IF, computed using k = 
1.75X10“V7, is also presented in Figure 11 as a 
function of Rm- An examination of the search situa- 
tion, using Figures 5, 6, and 11, shows that binocu- 
lars, even under the best of conditions, are not as 
effective as the naked eye if the meteorological vis- 
ibility is less than 10 nautical miles. 

As examples of the results to be obtained from 
Figures 5 and 11, values of W for various altitudes 
and meteorological visibilities are presented in Table 
1 for the three targets considered in Figure 5. To 
compute such tables the following procedure is em- 


Table 1. Sweep width W in nautical miles for naked-eye search. 


Altitude Meteorological Visibility (nautical miles) 


(feet) 

1 

3 

5 

10 

15 

20 

30 

40 

50 

500 

0.9 

3.2 

Submarines or small merchant vessels 

4.3 7.5 8.6 

9.6 

11 

12 

13 

1,000 

0.9 

3.7 

5.3 

8.5 

11 

12 

14 

15 

16 

2,000 


3.7 

5.9 

9.6 

12 

14 

17 

18 

19 

3,000 



6.4 

11 

13 

15 

18 

20 

21 

5,000 




12 

15 

17 

20 

22 

25 

500 

0.9 

3.7 

5.3 

Medium-sized ships 
12 

15 

17 

20 

22 

25 

1,000 

0.9 

3.7 

6.4 

13 

16 

19 

25 

27 

29 

2,000 


3.7 

7.0 

14 

18 

22 

28 

31 

34 

3,000 




14 

19 

24 

30 

34 

37 

5,000 




14 

21 

26 

33 

37 

42 

7,000 





21 

27 

34 

40 

45 

500 

0.9 

3.7 

Large combatants and high-speed liners 
6.4 13 16 

19 

25 

27 

29 

1,000 

0.9 

3.7 

6.4 

14 

18 

22 

28 

31 

34 

2,000 


3.7 

7.0 

15 

19 

25 

31 

36 

37 

3,000 



7.0 

15 

21 

26 

34 

40 

43 

5,000 




15 

22 

28 

36 

43 

48 

7,000 





22 

29 

38 

45 

50 

10,000 






30 

40 

47 

53 


SUBMARINE AND SURFACE SHIP SIGHTINGS FROM AIRCRAFT 


61 


ployed. For each pair of values of V and h, V /}f is 
computed and the corresponding value of Rm/h^ is 
obtained from Figure 5, from which is computed. 
Knowing R^^, W is obtained from Figure 11. 

The value of k obtained from the operational data 
provides means of determining the average effective- 
ness of search from aircraft. The value of k obtained 
from the Craik scanning experiments described in 
Section 4.4 was 2.73 X 10“^. Since this is for scan over 
45 degrees instead of 180 degrees it must be divided 
by four for comparison with the operational value 
of 1.75 X 10“^ This division gives 6.8 X 10~^, about 
four times the operational value. It is clear, there- 
fore, that the entire crew of an aircraft is about 
equivalent to one-quarter of an ideal lookout carry- 
ing out search under laboratory conditions. A num- 
ber of reasons for this discrepancy can be enumer- 
ated. These are soiled or scratched windows, fa- 
tigue, interruption from search caused by other 
duties, inability to search along an accurately de- 
termined best scanning line, and nonuniform angular 


distribution of scanning effort. Soiled or scratched 
windows tend to reduce the apparent contrast and 
hence both Rm and Oq. The need for keeping the 
windows clean and clear can not be overemphasized. 
On long sorties, fatigue is inevitable. However, its 
effects can be minimized by relieving monotony. This 
can be done by making frequent exchanges in station 
for the various lookouts. The best scanning line, 
(i.e., the locus of points on the ocean surface distant 
Rm from the aircraft) is usually 3 or 4 degrees below 
the horizon. A rough and ready rule for finding this 
locus is to extend the fist at arm’s length and look 
about two or three fingers below the horizon. There 
is good reason to believe that the scanning effort, 
instead of being uniformly distributed over the for- 
ward 180 degrees is heavily weighted in the front 45 
or 90 degrees. One suggestion for overcoming this 
tendency is to assign all lookouts, other than pilot 
and copilot, to the two 45 degree sectors just for- 
ward of the beams, leaving the forward 90 degrees 
to the pilot and copilot. 




Chapter 5 

RADAR DETECTION 


INTRODUCTION 

T he immense value and versatility of the radar- 
echo principle in its military applications were 
amply demonstrated during World War II. The abil- 
ity of radar to provide precise values of the range 
and bearing of objects on or above the surface of 
the sea, under all conditions of visibility, and fre- 
quently at distances considerably beyond the range 
of the human eye, assured its constant naval use 
as an instrument of search and early warning. In 
addition, it readily proved its usefulness in a number 
of related applicabilities. These included fire control, 
identification, altimetry, and aid to bombing, inter- 
ception, fighter direction, station keeping, and navi- 
gation. 

Nevertheless, airborne and shipborne search radar, 
used both offensively and defensively to gain contact 
with enemy forces, and to locate missing friendly 
units, represented perhaps the most widespread and 
successful of these military applications, and may be 
expected to continue to do so in the immediate future. 
It is with this aspect of radar that the present chap- 
ter deals. Emphasis is placed on basic search con- 
siderations; only those technical questions which 
have bearing on this subject are discussed. 

52 MODERN SEARCH RADAR 

CHARACTERISTICS 

The type of radar currently most useful in sea 
search is the airborne microwave (wavelength X ^ 
10 cm) search radar. A brief outline of the principles 
of operation of such equipment will serve in a gen- 
eral way to illustrate those of similar sets, including 
shipborne search gear. 

High-frequency radio energy generated in the 
transmitter of such a radar is led through a wave 
guide (a resonant copper pipe of rectangular cross 
section) to a funnel-shaped horn, or to a dipole 
radiator, located at the focus of a paraboloidal metal 
reflector. The energy is re-radiated from this re- 
flector in a lobe-shaped pattern, as indicated cross 
sectionally in Figure 1. The beam Avidth — conven- 
tionally defined as the angle d between half-power 


directions — is determined by the size of the paraboloid 
relative to X. In practice subsidiary back and side 
lobes (not shown in the figure) are produced in ad- 
dition to the main lobe; but these can be minimized 
by proper antenna design — i.e., by modification of the 
reflector shape, addition of parasitic radiators, etc. 


Figure 1. Idealized pattern of a microwave search 
radar. Length of radius vector is proportional to power 
radiated in that direction. 

Actually, in most search equipment, the paraboloidal 
reflector is truncated or otherwise modified in such 
a Avay as to produce a narrow antenna pattern in the 
horizontal plane (sometimes as narrow as 1 degree) and 
a relatively broad pattern (usually about 10 degrees) 
in a vertical plane. This provides extended coverage 
in altitude and lessened sensitivity to antenna tilt, 
features particularly desirable for airborne early 
warning and shipborne aircraft warning radar. 

In present radar sets, high-frequency energy is 
never emitted continuously; instead, it is trans- 
mitted in successive high-power pulses of very short 
duration. In order to decrease the minimum radar 
range, and to improve range resolution, as well as to 
permit the use of higher peak transmitted power, it 
is desirable to reduce the duration of such pulses to a 
minimum. However, against these considerations 
must be balanced the fact that a very sharp pulse 
wave form can be reproduced accurately only by a 
broad-band receiver; and an increase in bandwidth 
results in an increase of receiver noise level relative 
to echo strength, with a corresponding reduction in 
maximum radar range. Pulse durations used in prac- 
tice are generally between 0.1 and 2 microseconds, 
with 1 microsecond a common value. The rate at 
which such pulses are repeated is usually between 400 
and 1 ,600 pulses per second in current search equip- 
ment. Such repetition rates allow a sufficient time 
interval for energy in one transmitted pulse to travel 
out to a distant target and be reflected back to the 
radar before the next pulse is emitted. In search for 



62 


MODERN SEARCH RADAR CHARACTERISTICS 


63 


very far distant targets, lower pulse repetition rates 
may be necessary. 

Signals reflected from a target are picked up by 
the same antenna used to transmit them; they are 
then amplified in the radar receiver and presented 
as a response on an indicator. The kind of presenta- 
tion most frequently employed in modern search 
radars is that of the plan position indicator (PPI 


causes a bright spot, known as a “blip,’’ to appear 
on the fluorescent screen at a radial distance pro- 
portional to the elapsed time, and therefore to the 
range of the target. The persistence of the screen 
is sufficiently great that such an intensifiedjspot 
usually remains visible for several seconds. When the 
electron beam has swept to the outer edge of the 
scope, corresponding to maximum range obtainable 



Figure 2. Plan position indicator (PPI) presentation. Plane bearing radar was at an altitude of 20,000 
feet, directly over Boston. Cape Cod may be clearly discerned near the center of the photograph. 


scope). In this type of “intensity modulated” indica- 
tion, response is obtained on the face of a cathode- 
ray tube by means of variations in the intensity 
of a radially sweeping electron beam. At the instant 
an energy pulse is transmitted from the radar an- 
tenna, this beam begins to sweep out at a uniform 
rate from the center of the indicator. When the re- 
flected pulse from a target returns to the radar, the 
amplified energy is used to increase the volume of 
the electron stream impinging on the scope. This 


on the particular range scale employed, it reverts al- 
most instantaneously to the center of the scope, 
and another pulse is emitted. The angular position 
(azimuth) of the sweep trace on the scope is de- 
termined by the direction in which the antenna is 
momentarily pointing. 

In order to obtain area coverage, the radar an- 
tenna is rotated about a vertical axis. Various radars 
permit manual control, sector scan, or 360 degree 
scan. The last is most often employed in airborne 




64 


RADAR DETECTION 


search, the antenna being rotated at rates usually 
between 5 and 24 revolutions per minute (12 rpm is 
a common value). The use of this type of scan in con- 
junction with PPI indication, in which the radial 
sweep trace of the electron beam is made to rotate 
in S3mchronism vith the antenna, results in con- 
tinuous presentation of a plan map of the region sur- 
rounding the radar, as illustrated in Figure 2. Maxi- 
mum scanning rate is hmited by the necessity of re- 
ceding several successive pulses to produce a notice- 
able bhp (estimates of this minimum number of 
pulses vary from 4 to 10, depending on the type of 
screen used). It is thus related to the antenna beam 
vidth and the pulse repetition rate, which in turn 
depends on pulse duration and the maximum average 
power the transmitter tubes can handle; scanning 
rate is therefore related indirectly to target resolu- 
tion. In present search equipment, between 10 and 
100 pulses reach the target per scan, so that the 
maximum possible scanning rate is not generally at- 
tained. It may be noted that vith rapidly scanning, 
narrow-beam antennas some “scanning loss” may be 
experienced, owing to a turning of the antenna dur- 
ing the finite echoing time. The effect is usually small 
in practice. 

Other types of antenna scan and visual presenta- 
tion than those described above are occasionally em- 
ployed in search equipment. Narrow-beam antennas, 
for instance, may be scanned hehcally, or rocked; 
some of the newer airborne sets have their antennas 
gyro-stabilized in order to prevent distortion of the 
PPI map due to antenna tilt. An alternative method 
of intensity modulated presentation sometimes en- 
countered is that of the B scope, in which range is 
measured along a vertical and bearing along a hori- 
zontal axis; this necessarily results in some distortion, 
but provides greater resolution of nearby targets. It 
usually is combined vrth an antenna scan of 180 
degrees or less in the forw-ard direction. 

53 VISUAL AND RADAR SEARCH 

Having outlined the basic operative features of 
current search radar equipment, we shall find it in- 
structive at this point to compare the process of 
visual search, dealt with extensively in the previous 
chapter, vith that of radar search. 

It wiU be recalled that the eye, when searching 
systematically, tends to look in one direction for a 
short period of time (of the order of one second), 


during which several fixations occur. It then skips to 
a new fine of sight, frequently differing in direction 
from the pre\4ous fine by as much as 10 degrees. 
Since, during any single fixation, the eye can resolve 
distant objects only within an arc of about 1 degree, 
the distant coverage pattern for \isual search tends 
to be ragged; there is a considerable probability that 
small objects at long ranges will be passed over. At 
shorter ranges, on the other hand, there is a broad 
lobe of peripheral \ision in w hich prominent objects 
off the direct fine of sight are readily detected (see 
Figure 7B, Chapter 4). 

In contrast, radar scans continuously, without 
gaps in its coverage, and does so over a considerable 
range, at a rate of scanning usually considerably 
greater than that of the eye. A minimum radar range 
limitation is imposed in airborne search by the shape 
of the vertical antenna pattern and by the antenna 
tilt setting. Furthermore, there will in general be a 
near-by “sea return” area, in w^hich transmitted 
energy is reflected back to the radar from waves, 
with the result that an irregular bright patch is pre- 
sented in the center of the PPI scope (see Figures 
2 and 3). The extent of this patch increases with the 



Figure 3. Schematic illustration of vertical antenna 
pattern and sea return area for airborne early warning 
radar. 

altitude of the antenna and the roughness of the 
sea. Targets within the sea return area cannot readily 
be detected in most cases, although special circuits 
in newer sets offer some improvement in discrimina- 
tion. Maximum radar range is generally limited (ap- 
proximately) by the horizon, although under certain 
conditions of energy propagation considerably greater 
ranges may be obtained. This effect wdll be discussed 
in Section 5.4. 

The types of search coverage obtained by the eye 
and radar are illustrated quafitatively, by means of 
contours of constant detection probabifity, in Figures 
4 and 5. It should be understood that only the gen- 
eral shape of these patterns, presented for com- 
parison, is significant. These diagrams serve to iUus- 


PROPAGATION 


65 


trate the fact that radar and visual search, as regards 
coverage, are to a large extent complementary. This 
is true in many other respects, too. While the eye, 
for instance, is .not directly capable of exact range 
determination, the accuracy of electronic timing cir- 
cuits makes radar well adapted for this purpose; on 




Figure 4. Schematic horizontal coverage patterns for 
(A) the eye, and (B) radar. Contours of constant prob- 
ability of detection are shown. 

the other hand, the eye possesses obvious superiority 
in target resolution. We may also note that, since 
radar wavelengths are large in comparison with the 
dimensions of smoke, dust, and water particles in the 
atmosphere, radar radiation (except that of ex- 
tremely high frequency) is only shghtly affected by 




Figure 5. Coverage obtained in scanning through 90° 
for (A) the eye, and (B) radar. 

such obstacles, whereas visible radiation is strongly 
attenuated or reflected by them. 

In the visual case, fixation on a target has been 
regarded as making detection certain; the element of 
chance is introduced by the uncertainty of obtaining 
a fixation. It is an important differentiating char- 
acteristic of search radar that, even vdth a well- 


tuned set scanning uniformly over a (small) target 
within easy range (a case analogous to assured fixa- 
tion) bhps may not be returned on every scan. This 
effect, largely a consequence of varying target as- 
pect, will be discussed at greater length in Section 
5.5. The point of chief^importance to the present 
discussion is that in both cases detection is a matter 
of uncertainty. Clearly, the probabihty methods de- 
veloped in Chapter 2 in the treatment of the general 
theory of detection will have apphcations in the 
sohing of radar detection problems. We shall in- 
vestigate the mathematical formulation of some of 
these in Section 5.6. 


54 PROPAGATION 

Radar energy is propagated in free space accord- 
ing to an inverse square variation with distance. That 
is, if Pt is the powder transmitted in a pulse, Gt the 
power gain of the antenna for transmission (a func- 
tion of its shape and size), and Po' the power density 
at distance r along the axis of the antenna. 


When the energy strikes a target at range r, a 
certain portion of it is scattered, that is, diffusely 
reflected (re-radiated). The scattering ability of the 
target is measured by a quantity <r, the “effective 
radar cross section,” defined as the ratio of total 
power reflected from the target to incident power 
density impinging from the direction of the radar. 
The amount of powder re-radiated from the target 
is therefore PoV. This powder is likewise attenuated 
according to the inverse square law; its density at 
the radar antenna is 

/ ^ PnV _ Pi Gt(T ^ /2) 

47rr“ (4x)V‘* 

The total powder received from the reflected pulse 
is the product of P^' and Gr, the gain of the antenna 
for reception, i.e.. 


In case the antenna is a paraboloid of aperture 
area A, it can be showm that the above expression 
reduces to 

p, = (4) 

47rr^X^ 


66 


RADAR DETECTION 


If -Pmin is the minimum value of reflected power 
that can be amphfied by the radar receiver to furnish 
a recognizable blip, we find, solving equation (4) for 
r (^\ith Pr = Pmin), that the maximum range of de- 
tection of a target of effective cross section a is 


^max 


'4 


GtGrO- 

■(4x)2 



( 5 ) 


Although these formulas have been developed for 
the case of propagation in free space, it is found that 
they may be employed to a reasonable degree of ap- 
proximation under many conditions encountered in 
practice. For example, in the case of sea search by 


airborne radar at common altitudes (such that the 
effects of interference vdth power reflected from the 
sea are of secondary importance), the last formula 
provides at least a quahtative indication of the in- 
fluence of various factors on maximum radar range. 
In particular, the relative insensitivity of rmax to 
large variations in transmitted power is shown : owing 
to the fourth root relationship, an increase in trans- 
mitted power of sixteen fold is necessary to double the 
maximum range. (In this connection, it might be 
pointed out that in Figure 1 of Section 5.2 of this 
chapter the reflected signal has sufficient strength 
to give good range over a wider beam than the power 
pattern would indicate. Thus, at the half-power 



Figure 6. Variation of echo amplitude with constant radar performance. Target 30 miles from radar, with path 
of pulses passing over sea. 


BLIP-SCAN RATIO 


67 


points, maximum range is reduced by a factor of only 
\lV2 = 0.84.) 

Near the surface of the sea, a different type of 
propagation is.. commonly observed. Energy that 
strikes the surface at a small angle is reflected, as 
from a good conductor, undergoing a 180 degree 
change of phase. Over a calm ocean, the alternate re- 
inforcements and cancellations of radar energy fol- 
lowing the two possible paths, direct and reflected, 
cause the antenna pattern to become stratified. With 
longer wave radars, the null regions thus produced 
cause a pronounced recurrent fading of echoes from 
airborne targets. But for the microwave sets in in- 
creasingly common use in search, the stratified lobes 
are so closely spaced (spacing less than 1 meter) 
that such fading ceases to be a problem. Under these 
conditions of transmission near a calm sea, the echo 
power received from a target also near the sea can 
be shown to vary according to an inverse eighth 
power of range, rather than an inverse fourth power. 
Near the surface of a rough sea, some intermediate 
law may, in effect, be more closely followed. 

As previously mentioned, maximum radar range, 
except in the case of very small targets, usually is 
not limited as a result of attenuation suffered by the 
radiated energy (for, although this attenuation is, as 
we have seen, considerable, correspondingly large 
amounts of power can be transmitted in the radar 
pulses), but is limited by the horizon. Actually, owing 
to refraction in the earth’s atmosphere, a portion of 
the energy is bent around somewhat beyond the 
horizon. It has been found that this effect can be 
taken into account approximately by computing 
maximum range as horizon range for a fictitious earth 
of radius four-thirds that of the real one. A con- 
venient formula based on this assumption is = 1.25 
(V% + where R is the maximum radar range 

in nautical miles, hr is the radar altitude in feet, and 
ht is the target altitude in feet. 

Under certain meteorological conditions, generally 
associated with inversion of the normal temperature 
or humidity gradients, abnormally long radar ranges 
may be observed. The effect, particularly noticeable 
if both radar and target are close to the surface of 
the sea, is much the same as if a portion of the 
radiated energy were trapped beneath the inversion 
level. Apparently, trapping of some energy in a sur- 
face duct does not in general interfere with propaga- 
tion at higher altitudes, since abnormally long de- 
tection ranges near the surface are frequently ob- 
served to be accompanied by good ranges from sur- 


face to air, or vice versa. In extreme cases, the exis- 
tence of anomalous conditions, both of abnormal and 
of subnormal propagation, may lead to very pro- 
nounced and rather rapid variations in echo strength, 
as indicated, for example, in Figure 6. (Note that on 
one of the days covered in this chart a variation in 
received power of 53 db, or 200,000-fold, in a period 
of 2}/2 hours is recorded.) In certain geographical 
regions, notably, the eastern seaboard of the United 
States, conditions of ‘‘anomalous” propagation are 
more or less prevalent. In most locahties, however, 
large or rapid fluctuations in propagation conditions 
are not generally to be expected. 

BLIP-SCAN RATIO 

It was mentioned in Section 5.3 that when a radar 
scans across a target, particularly near the limit of 
its range capabilities for that target, it is the general 
experience that a blip is not presented on the radar 
indicator on each scan. In order to characterize 
analytically the behavior of a given radar with re- 
spect to a specified target, it is convenient to intro- 
duce the concept of “blip-scan ratio.” This ratio, 
which we shall denote by is defined as the pro- 
portion of scans, upon a target at range r, during 
which a recognizable signal is presented on the PPI 
scope. It therefore represents the probability that a 
single scan will produce an effective blip, i.e., a blip 
which is actually recognized by an operator who is 
focusing his attention on the part of the scope where 
it appears: a “recognizable” blip may actually fail 
to be recognized by an operator whose attention is 
lagging, or is directed to another part of the scope 
where objects of interest are seen; this effect of 
operator fallibility will be considered later. In other 
words, we are separating the study of the uncertain- 
ties (probability) of radar detection into two parts: 
the question of the probability \{/ of detection of the 
blip when the operator is concentrating on the part 
of the scope where it occurs, and the matter of how 
likely he is to be so concentrating. While this separa- 
tion is somewhat unrealistic (an intense bhp is apt 
to be seen out of the corner of the eye, a faint but 
“recognizable” one is not), it affords a convenient 
simplification and will be made the basis of the 
present treatment. 

Clearly, the value of the blip-scan ratio is de- 
pendent on a number of long-term variables that 
may in an approximate treatment be regarded as 


68 


RADAR DETECTION 


constant during a particular search, or at least dur- 
ing long parts thereof. These include type of target, 
conditions of propagation, sea state, direction of 
search with respect to that of the wind, radar alti- 
tude, antenna tilt, level of operator and set per- 
formance, and radar characteristics such as wave- 
length, scanning rate, etc. In addition, two basic 
short-term variables are involved — range and target 
aspect. As we shall see, the aspects of naval targets 
vary characteristically in short-time cycles; it is this 
fact, represented by appropriate mathematical as- 
sumptions, that permits the treatment of ^ as a 
specific function of range only. Before formulating 
these assumptions, however, let us investigate briefly 
the general subject of radar echo fluctuations. 

The components of radar targets which are most 
effective in reflecting energy are flat surfaces (normal 
to the axis of the radar beam) and internal rectangu- 
lar corners. Metallic conductors are always more 
effective than nonconducting materials. Since a flat 
surface reflects radiation specularly, its orientation 
must be within a few degrees of normality to the 
beam direction to return an appreciable signal; other- 
wise, most of the incident energy is shunted off into 
space. The corner reflector, on the other hand, has 
the property for microwave radar energy, as for light, 
of reflecting radiation along the direction of in- 
cidence, over a wide range of angular aspect. It is 
thus relatively insensitive to momentary aspect and 
movement. 

We may accordingly in a general way distinguish 
two types of naval targets. The first type, which we 
shall call Class A, represented by large vessels, in- 
cluding warships and merchant ships, and in most 
cases by beam-aspect surfaced submarines, is char- 
acterized by the prevalence of flat reflecting surfaces 
and rectangular corners and brackets. Energy re- 
flections from the many components of such targets 
reinforce or cancel in accordance with their various 
momentary phase relationships. When micro- wave- 
lengths are employed, very slight target movements 
cause radical alterations in these phase relationships, 
with consequent rapid variations in echo strength. 
In the case of naval targets, such movements may be 
represented by ship roll and pitch, or even by physi- 
cal distortion of the vessel in a seaway, bending of 
the masts in a wind, etc. It is found for targets of this 
type that the echo fluctuations are rapid, even in 
comparison with the short time interval required for 
the beam of a searching radar to sweep across them. 
Consequently, on any particular scan, although large 


momentary (pulse-to-pulse) echo variations occur, 
an average signal strength, resulting from summa- 
tion of the pulses, is presented on the radar indicator 
at a roughly constant level. Since this average level 
does not vary greatly from scan to scan, but in- 
creases steadily with decreasing target range, the 
blip-scan ratio for targets of Class A usually in- 
creases from zero to unity over a rather short range 
interval. It is apparent that this corresponds ap- 
proximately to a “definite range law,” the definite 
range being that at which the average signal first 
becomes perceptible. The effect is illustrated in 
Figure 7. 



Figure 7. Blip-scan ratio for Class A and Class B 
targets. 


The second type of radar target, which we shall 
call Class B, consists of relatively smooth, con- 
tinuously curved surfaces, having few sharp corners 
or angles. Examples are surfaced submarines in bow 
or stern aspect, periscopes, and schnorchels (cylin- 
drical submarine “breathing” mechanisms). The 
energy return from such targets also varies with their 
momentary aspect, but the fluctuations are much 
slower than those observed for Class A targets. In- 
deed, the rate of variation in this case, being usually 
of the order of magnitude of the scanning rate, is 
sufficiently slow that the energy return per pulse does 
not, in general, change radically during the time the 
radar beam is sweeping across the target. From scan 
to scan, however, there is wide variation in signal 
intensity. As illustrated in Figure 7, the blip-scan 
ratio for such small, smoothly shaped targets is in 
no way suggestive of a definite range law. 

Although most targets fall into one or the other 
of the classifications described above, a few, such as 
intermediate-aspect surfaced submarines, cannot 
readily be regarded as either large, complex targets 
or small, simple ones. It is convenient therefore to 
extend our definitions of these classes: Class A will 


BLIP-SCAN RATIO 


69 


include those targets for which, to the desired degree 
of approximation (with specified values of the long- 
term variables) a definite range law can be defined; 
Class B will include all others. Methods of dealing 
with the definite range situation, including averaging 
for distributions of parameters, may best be illus- 
trated in connection with sonar search, which forms 
the subject of Chapter 6. For the remainder of the 
present study, therefore, we shall confine ourselves 
largely to the study of Class B targets. 

We are now ready to formulate one of the assump- 
tions, previously mentioned, regarding variations of 
target aspect. We shall assume that they are of such 
a nature that the blips returned from a target at a 
certain range are distributed at random among the 
radar scans (with a relative frequency corresponding 
to the blip-scan ratio at that range), in other words, 
that xpir), the probability that a blip be presented 
on a particular scan, is independent of what may be 
known about the results of previous scans. This as- 
sumption, as we have shown, represents a reasonable 
approximation to operational facts, for most targets 
and for common scanning rates. It greatly simplifies 
the mathematical treatment of radar detection prob- 
lems, to be dealt with in the following section. 

The effect of changes in general target aspect, re- 
sulting from relative travel, has been found to be of 
secondary importance. For symmetrical targets such 
as schnorchel, this is obviously the case. And al- 
though, for other targets, unusually long ranges can 
be obtained at certain general aspects, the angular 
regions, in the horizontal plane, over which this is 
true are found in practice to be relatively small. 
(The situation is indicated schematically in Figure 
8, for several types of targets.) We are therefore 
justified, in most cases, in assuming that these angu- 
lar regions may be ignored and that the blip-scan 
ratio may be regarded as approximately independent 
of general horizontal aspect. 

The problem of predicting blip-scan ratio from 
theory would be found (if it had to be dealt with) 
to be a formidable one, indeed. Not only would it 
be necessary to make computations of effective radar 
cross section o- — dependent on the size, shape, and 
materials of the target, and on the wavelength and 
polarization of the radar radiation— but, in addition, 
assumptions as to the distribution of values of a with 
respect to time would be required. A theory involv- 
ing such variables and assumptions would necessarily 
tend to become overcomplicated and artificial. 
(Nevertheless, efforts made to show with sufficiently 


simple geometrical figures, such as cylinders, that the 
blip-scan ratio should vary with sea state in the 
observed manner have met with some success; their 
extension to more complicated figures, however, 
would be prohibitively difficult.) It is fortunate, 
therefore, that we are not forced to rely upon theory 
for our knowledge of blip-scan ratio. Test data are 
available, in considerable abundance, which provide 
reasonably accurate values of \p{r) for specified sets 





Figure 8. Typical polar diagrams, showing contours of 
constant detection probability as a function of general 
horizontal aspect for various kinds of targets. 

of values of the long-term variables, and, as a prac- 
tical matter, it is this availability that makes the 
concept of blip-scan ratio useful. 

The effect on \f/{r) of one long-term variable, the 
type of target, has been indicated in Figure 7. In 
Figure 9 the effects of two others, sea state and di- 
rection of search relative to that of the wind (up- 
wind and downwind search) are illustrated for 
AN/APS-15A radar used against schnorchel tar- 


70 


RADAR DETECTION 


get. The operational significance of variations in 
these and other basic parameters may perhaps be 


1.0 
0.8 
_ 0.6 
0.4 

0.2 

0 

0 2 4 6 8 10 12 14 16 

RANGE IN NAUTICAL MILES 

Figure 9. Effect of sea state and wind direction on 
blip-scan ratio for AN /APS- 15 A, altitude 500 feet, 
schnorchel target (experimental data from ASDevLant 
Project 547). 

better understood in terms of their influence on radar 
search width, to be discussed in Sections 5.6 and 5.7. 



56 RANGE DISTRIBUTIONS AND SEARCH 
WIDTH 

We shall now study the application to radar de- 
tection problems of the methods of analysis outfined 
in general terms in Chapter 2 for the determination 
of sighting range distributions and search width. The 
coordinate system (Figure 7, Chapter 2) and much 
of the notation previously employed will be retained; 
in particular, it will be recalled that w is the relative 
speed of target and searching craft, T the glimpsing 
(radar scanning) period and g(\/x^ + 2/^) 
^‘glimpse’’ probability of detecting on a particular 
scan a previously undetected target located in the 
neighborhood of the point ix,y). 

In Section 2.5 it was shown that the probability 
of first detecting in the area dxdy a target that has 
moved along in relative space parallel to the y axis 
to the neighborhood of the point {x,y) is 

( 6 ) 

and that the average detection rate in dxdy for unit 
target density is 

p{x,y) = (7) 

In both cases, 

00 

= - ;^^log|^l - + {y - iiyT)2^J,(8) 


the summation extending to infinity because the rela- 
tive track is assumed to be of indefinitely great ex- 
tent in the direction toward which the searcher is 
traveling. The distribution of true ranges of first de- 
tection is expressed [see equation (35), Chapter 2] by 

X TT +0 

rp(r sinf, r cos^ (9) 

-6 

the integration extending over the angular scanning 
range, assumed symmetrical about the y axis {d rep- 
resents the radian measure of half this range; for 
all-around scanning B = tt). The function p(r) may 
be alternatively regarded as the rate of detection in 
a unit range interval at range r for unit target 
density. If, as is the case in certain operational tests, 
the searching craft makes its approach to the target 
on a radial course, the sighting range distribution is 
expressed by equation (7) for lateral range zero, i.e., 

p(r) = e-«o..) ffM. (10) 

The quantity wT that appears in equation (8) rep- 
resents the distance traveled by the target along its 
relative track between successive radar scans. Since 
it is small in most cases {wT = 0.21 nautical miles, 
for a 150-knot searching aircraft, stationary target, 
and 12 rpm scanning rate), an important simplifica- 
tion of the expression for F(x,y) can be made (the 
summation can be replaced by integration). Thus, 

F(x,y} = “ g(Vx^ + y^)]dy, (11) 

the upper limit of integration lying in the scanned 
range of ^ = a; cot f, for f in the range (tt — 0, tt + 0). 

Writing r(r) for the quantity —log [1 — g{r)], 
equation (9) becomes 

pir) = [“ 

By analogy with the corresponding equations of 
Chapter 2, the distribution of lateral ranges of first 
sighting is given by 

[ 1 /•Ixl cot (jr-0) “I 

- r(r)d?/J; (13) 

the radar search width W, corresponding to the area 
under the lateral range curve, is 

noo [ PI f\x\ cot {-K — 6) “I'l 

^ = LS - r(.)rf,J}dx; 

(14) 


RANGE DISTRIBUTIONS AND SEARCH WIDTH 


71 


and the average detection range is 

f 

r(r) can readily be expanded, 
r(r) = - log [1 - g(r)] 

= + i gKr) + J g\r) + • • • . (16) 

Under the condition 

gir) < < 1, (17) 

r(r) may be replaced by g{r) in equations (12) to 
(15), thus completing the analogy (writing 7 for g, 
1/w for IfwT, and tt for 6) with equations (1), (3), 
(31), and (36) of Chapter 2, derived for the case 
of continuous all-around looking. Although condi- 
tion (17) is sometimes not satisfied for all ranges, it 
is often satisfied at ranges of particular interest. 
Thus, in the calculation of lateral range distributions 
for the larger targets, it is found that p{x) approaches 
unity while g{r) is still small. The approximation 

m = g{r) ( 18 ) 

may therefore, if used judiciously, yield much useful 
information and shorten computation considerably. 

In order to obtain an analytical expression for g(r), 
the single-scan probability of detecting a target at 
range r, it is necessary to determine the relationship 
between this detection probability and the blip-scan 
ratio (Section 5.5), representing the single-scan prob- 
ability of obtaining a recognizable blip on the radar 
indicator. We shall discuss in detail a particular set 
of assumptions regarding this relationship, and also 
consider briefly the effects of slight alterations in 
these assumptions. 

Concerning the radar operator, we shall make the 
assumption that if he has seen no blip on a particular 
scan his probability of noticing a recognizable signal 
on the succeeding scan is po; and that if he has seen 
a blip he will be alert on the following scan, certain 
to detect any blip presented. If, however, no blip is 
presented on this second scan, the operator is as- 
sumed to lose interest, his chance of noticing a new 
signal reverting to po. The value of this probability 
is dependent on the state of training and fatigue of 
the operator; we shall, for the time being, regard it 
as a known constant and as independent from scan 
to scan. We shall also assume that the operator must 
see a certain number n of successive blips for detec- 
tion of a target to occur. For rapidly scanning air- 


borne search radars, experience indicates that n has 
the value 2 or 3 in most cases. 

Under these assumptions, we see that a necessary 
and sufficient condition for the detection of a target 
on a particular (fth) scan, given that no previous de- 
tection has occurred, is that bhps be returned on 
that scan and on the preceding (n — 1) scans, and 
also that the operator see the first of these n suc- 
cessive blips. Expressed in symbols, 

gi{r) = po n ^ {y - iwTf). (19) 

j=i-n+l 

If icT is small, if n is small, and if is a slowly 
varying function of range (all of these conditions are 
generally satisfied in practice), then the t/'’s in the 
above expression are all nearly equal. Therefore, 
dropping the subscript, we have approximately 

g{r) = poVir). (20) 

Note that the multiplication of the xp’s in equa- 
tions (19) and (20) is permissible, from the prob- 
ability viewpoint, only if the independent probability 
assumption mentioned in Section 5.5 is justified. 

Condition (17) under which r(r) may be replaced 
by g{r) becomes 

PqV(x) < < 1. (21) 

It is satisfied at ranges for which the blip-scan 
ratio is small, and at all ranges if the operator is very 
inattentive. It should be noted that this “inatten- 
tiveness” may be of an effective kind, the result not 
only of actual distractions and fatigue but also of 
difficulty in finding the target blip among others of 
a random nature, dependent on noise level and char- 
acteristics of presentation. The value of po will there- 
fore in operations frequently be very small. 

These assumptions regarding operator efficiency 
and the criterion of detection have been chosen be- 
cause of their reasonable nature and the simplicity 
of the result they yield. They are by no means the 
only ones that might be made, and are, indeed, im- 
mediately suggestive of several similar ones, of per- 
haps equal validity. For instance, it may be that the 
detection requirement, instead of the occurrence of 
n successive blips, is the occurrence of n blips dis- 
tributed in any way among s scans (s being a small 
number, greater than n). In this case, treating \l/ as 
constant during the s scans, we have 

g(r) = PoC:tA”(l - (22) 

= PoCniA” + higher terms. 


72 


RADAR DETECTION 


where Ch is the binomial coefficient 

s! 

"" n! {S-ny: 

If \J/ is sufficiently small that the higher terms of 
equation (22) can be neglected, we again have an 
expression for g(r) of the same form as that of equa- 
tion (20), differing from it only by a constant factor. 
We conclude that for small values of yp — those, as 
we have pointed out, which are usually of greatest 
interest — this form of g{r) is insensitive to the exact 
nature of the assumptions made in determining it. 
We shall therefore use as our radar detection law 

g{r) = hV{r), (23) 

in computing range distributions and sweep width 
from equations (12) to (15). The exact value of the 
constant k (as in the analogous visual sighting case) 
and also of n, is best determined by comparison with 
test (or, for some purposes, operational) results. 

57 COMPUTATIONAL METHODS 

The application of the formulas derived in the 
preceding section to the solution of specific radar 
detection problems may be illustrated by a brief 
discussion of computational methods. In particular, 
we shall be interested in the utilization of test data 
for the predicting of operational results and in 



0 2 4 6 8 10 12 14 16 


RANGE IN NAUTICAL MILES 

Figure 10. Graph of F (r) = log [1 as a func- 

tion of range, using blip-scan data of Figure 9. Com- 
puted as an intermediate step in the determination of 
the true range distribution for search ahead only. 

methods of correlating theoretical results with those 
of past operations. 

The distribution of detection ranges under a par- 
ticular set of conditions for search ahead only (zero 
lateral range), characteristic of certain convenient 
test procedures, may readily be computed by means 


of a single integration for each range value, provided 
blip-scan data obtained from tests made under the 
specified conditions are available; and provided as- 
sumptions are made regarding the values of the con- 
stant k and the number n of recognized blips neces- 
sary for detection. The integration to obtain F{0,r) 
is performed graphically, in accordance with equation 
(11), first plotting r(r) against r (see Figure 10). Use 
can be made, if desired, of the simplification em- 



Figure 11. True range distribution for search ahead 
only. Computed using blip-scan data of Figure 9. 


bodied in equation (18), for most values of r. The 
true range distribution is then computed directly 
from equation (10). Figure 11, as an example, illus- 
trates the results obtained using the blip-scan ratios 
of Figure 9 and the assumptions k = 1 and n = 2 
[i.e., g{r) = \p'^{r)]. The particular value of such cal- 
culations is that range distributions computed under 
various assumptions as to the values of k and n 
may be compared with the range distributions actu- 
ally obtained in the tests (provided these are sta- 
tistically significant), as a means of determining 
which of these assumptions provide the best fit. Such 
trial-and-error calculations offer the most practical 
method of evaluating these parameters. It should be 
noted, however, that parameter estimates based on 
tests in general give optimistic results, as compared 
with those of operations; it may therefore be pref- 
erable, when such are available, to employ opera- 
tional (rather than test) data, in the manner out- 
lined in Section 4.10. 

The computations involved in determining true 
range distributions for the case of sector or all- 
around search for targets uniformly distributed at 
all lateral ranges are similar in principle to those out- 
lined above, but' are more complicated, since addi- 
tional integrations are required. The method of pro- 
cedure, in brief, is to express equation (11) in polar 


COMPUTATIONAL METHODS 


73 


coordinates by means of the familiar relationships 
X = r sin f , 2 / = r cos f ; then, for fixed values of f 
and of r, to determine by graphical integration the 
value of this expression and consequently of the 
integrand of equation (12); finally, for each value of 
r, to perform graphically the f integration required 
by the latter equation. If this is done for enough 
values of r, the desired range distribution may be 
determined as accurately as required. Figure 12 illus- 



Figure 12. True range distribution for all-around 
search. Computed using blip-scan data of Figure 9. 


trates the type of distribution obtained. As before, 
AN/APS-15A blip-scan data are supplied by Figure 
9, and the assumptions = 1, n = 2 are made. As 
regards the general shape of the distributions, the 
results are seen to be not far different from those 
obtained for the previous case of search straight 
ahead. Sector scanning radars (such as AN/APS-3, 
which scans 150 degrees forward) also give true range 
distributions of a similar character. 



Figure 13. Lateral range distribution. Computed us- 
ing blip-scan data of Figure 9. 


The distribution of lateral detection ranges is most 
easily computed in terms of rectangular coordinates. 
Equation (11) is evaluated much as before, with the 


integration extending over all values of y in the 
scanned region, for fixed values of lateral range x. 
Equation (13) then gives the corresponding ordinates 
of the lateral range distribution. Graphical integra- 
tion of the curve so obtained yields the value of the 
search width W [equation (14)] directly. Figure 13 
shows lateral range curves obtained under the same 
assumptions as before; corresponding values of sweep 
width are indicated. 

By collecting test data on ^(r) for known sets of 
conditions, varying only a single parameter, we can 
readily determine the influence of this parameter on 
search width. As an example, the effect of aircraft 
altitude on W is shown in Figure 14, for the case of 



0 500 1000 1500 2000 


ALTITUDE IN FEET 

Figure 14. Search width IF as a function of aircraft 
altitude. Search for surfaced submarine by AN/ASP- 
15 A, sea states 1 and 2. 

search for surfaced submarines, with the usual as- 
sumptions, A; = 1, n = 2. The differences in search 
width for beam and bow-stern runs against this type 
of target are clearly shown. As previously indicated, 
however, true beam runs are encountered in opera- 
tions with relative infrequency, so that the results 
for bow-stern runs give a truer average picture. 

The effect on W of variations in another param- 
eter, direction of search with respect to that of the 
wind, has been indicated roughly in Figure 13, for 
search against schnorchel in sea states 1 and 3. In 
sea state 1, it will be observed, wind direction is un- 
important; in sea state 3, however, upwind search 
is already impossible, and the search width is greatly 
reduced even for downwind search. Results of an 
intermediate character are obtained for sea state 2. 
If area search is to be conducted in sea states greater 
than 1 by means of parallel sweeps in one direction 
(with a number of searching craft), it is clearly most 
advantageous to choose that direction as downwind. 
Usually, however, search is conducted on a round- 



74 


RADAR DETECTION 


trip or shuttle basis; if this is the case, it might prove 
best to search at an angle to the wind. (Confirmation 
of this conjecture, however, awaits the obtaining of 
further test data on blip-scan ratio for crosswind 
search.) It should be noted that these remarks apply 
only to very small Class B targets, such as periscopes 
and schnorchels. For larger targets, including sur- 
faced submarines, differences between upwind and 
downwind search are usually found to be of little 
significance. A useful empirical relationship regarding 
search width for these larger targets is that W is 
roughly equal to twice the range at which — 0.1, 
xl/{r) being approximately independent of wind direc- 
tion. (Since the blip-scan ratio for such targets rises 
rapidly in this range, W is not sensitive to the exact 
value xp = 0.1.) 

We have thus far dealt only with situations such 
as those to be encountered in future operations, in 
which conditions may be regarded as homogeneous, 
i.e., situations in which specific values may be as- 
signed to each of the parameters in our equations. If 
we desire that our theory duplicate actual past op- 
erational results, however, the analysis does not pro- 
ceed so simply. In this case it is necessary to adopt 
methods of averaging for distributions of the basic 
parameters. Occasionally these distributions are 
known; more frequently they must be estimated. 
The reader is referred once more to the final para- 
graphs of Chapter 2, in which the topic of long- and 
short-term parameter variations in relation to the 
analysis of past and future operations is summarized. 

Owing to the inadequacies of available operational 
data and of knowledge concerning distributions, we 
shall not here enter into the discussion of averaging 
methods, but defer this until the next chapter. We 
may, however, mention again, for emphasis, a few 
of the long-term factors, the variations of which are 
particularly significant in the analysis of past op- 
erations. Namely, (1) set performance, which may 
change slowly over considerable periods of time (as 
in the case of seepage of moisture into wave guides 
or radomes, causing a gradual decline in the per- 
formance of certain types of radar) or more rapidly 
with changes in set tuning; (2) propagation condi- 


tions, which as we have pointed out may vary rather 
radically and unpredictably in certain localities ; and 
(3) operator performance, which depends on train- 
ing, experience, and alertness. Qualitatively, the cumu- 
lative effect of these factors is always to increase con- 
siderably the dispersion of operational range distribu- 
tions, and usually to reduce average detection ranges 
and search widths (sometimes by a factor of 2 or 3). 
It is expected that analyses of operational data on 
Class A targets, for which the dispersion is largely 
due to such unassessed factors, will lead to a better 
understanding of these effects. Figure 15 shows an 
example of the type of range distribution obtained 



RANGE IN NAUTICAL MILES 


Figure 15. Operational true range distribution. Ob- 
served ranges of radar first contacts on surfaced sub- 
marines leading to attacks for the period July 1943 to 
March 1944. (Includes all types of radar, day and night 
service, under all weather conditions.) 

in operations, during the earlier part of the past war, 
for surfaced submarines. 

We have omitted consideration of a number of 
important topics relating to radar search, such as, 
for example, the effects of forestalling due to counter- 
measures (search receivers) in search for enemy units. 
Such forestalling, might possibly be met by operat- 
ing the searching radar intermittently; but the effect 
of such operation on the radar search width must 
also be taken into consideration. We have likewise 
omitted discussion of visual forestalling and the cal- 
culation of combined radar and visual search widths 
(necessary for the determination of optimum search 
altitude). Such topics represent in themselves sepa- 
rate subjects, which must await more detailed study 
elsewhere. 


Chapter 6 

SONAR DETECTION 


SONAR SEARCH— GENERAL 

I N SUBSURFACE WARFARE, reliance must be placed 
on sound (or supersonics) for search and detection, 
since sea water is virtually opaque to electromagnetic 
waves. Neither visual nor radar search is possible in 
a water medium. Consequently sonar must be used 
when searching for submerged submarines, torpedoes, 
mines, or other underwater objects. Magnetic detec- 
tion is also possible, but the range of detection is 
normally much less than for sonar. Sonar search, 
therefore, is of importance in the operation of sub- 
marines and of antisubmarine forces. 

Sonar detection involves either listening or echo 
ranging. Simple listening gear consists of a receiver 
and amplifier which pick up sounds generated by the 
target and present them to the sonar operator’s ear. 
Echo-ranging gear has a transmitter in addition, 
which sends sound into the water; the sound received 
is then an echo reflected from the target. In general, 
listening gear has the advantages of simplicity and 
long detection range on a noisy target, but is not 
effective if the target runs quietly. Echo-ranging 
search has the advantage that it cannot be defeated 
by slow-speed quiet running. In addition, echo ranging 
provides information on range to the target, which 
hstening does not. As a result, both listening and 
echo ranging are used for search, with preference 
sometimes given to the former, sometimes the latter. 

In this chapter examples of both listening and 
echo-ranging search will be discussed, but only as 
examples to illustrate the type of problems involved. 
The aim of the operational analysis of sonar search 
is to determine how search gear or searching craft 
should be used in any particular situation to give 
the best result. This result may be expressed as a 
lateral range curve or a sweep rate in accordance 
with Chapter 2. There are a great many factors which 
determine the lateral range curve, involving char- 


acteristics of the gear, its operation, the target, its 
behavior, and sound transmission in the ocean. Some 
variables, such as speed of searching ship, can be 
determined by the searcher, whereas others, such as 
sound conditions, are beyond his control. The values 
of these uncontrollable variables often determine 
how the values of the others should be chosen. In 
any particular case, the various factors must be con- 
sidered in detail, and the lateral range curve obtained 
in accordance with estimates of the physical situa- 
tion. Before analyzing any typical problems, how- 
ever, it is worth while giving a general outline of the 
factors which come into play. 

In detecting the target the sonar operator must 
distinguish the signal from the everpresent back- 
ground noise. Hence the factors of interest can be 
divided into the three following classes: 

1. Those which influence the strength of the signal 
which it is desired to detect, the signal being either 
noise incidental to the operation of the target, sound 
transmissions by the target, or an echo reflected 
from it. These include characteristics of the sonar 
gear being used, of the target, and of the ocean. 

2. Those which determine the background level 
against which the signal must he heard, including noise 
from own ship, noise from waves and animal life 
in the ocean, and, in the case of echo ranging, re- 
verberation. 

3. Psychological factors and characteristics of the 
sonar data presentation which determine the prob- 
ability of detecting a given signal in the presence of 
a given background. Each of the subdivisions must 
be considered for both listening and echo ranging. 
They are exhibited in parallel columns, echo ranging 
on the left, listening on the right. Corresponding 
topics appear side by side, and when identical con- 
siderations apply to both cases, their treatment is 
written across the columns (i.e., the column division 
is temporarily abandoned). 








75 


76 


SONAR DETECTION 


ECHO RANGING 


LISTENING 


The scheme of echo-ranging detection is shown in 
Figure lA: 


The scheme of listening detection is shown in 
Figure IB: 




Figure lA. Two-way sound transmission. 


Figure IB. One-way sound transmission. 


Factors Influencing Signal Strength 

1. Intensity of transmitted pulse: 

The intensity of the transmitted pulse at a given 
range (say 1 yard) depends on the acoustic power 
output of the gear, and its directionality, the more 
directional the transmitter the greater the intensity 
for a given total power output. Standard echo-rang- 
ing gear now used by antisubmarine ships has an 
intensity of about 182 decibels above 0.0002 dynes 
per square centimeter at 1 yard on the axis of the 
projector. 


1. Sound output of target: 

The sound output of a ship depends primarily on 
the type of ship and its speed. At very low speeds 
machinery noise often predominates, which is al- 
most entirely low-frequency sound. Propeller cavita- 
tion noise containing the higher sonic and low super- 
sonic frequencies becomes important at normal 
speeds. A submerged submarine, however, produces 
this cavitation noise less readily the deeper it sub- 
merges. In addition, individual ships vary consider- 
ably from the average performance, especially in the 
details of their sound output. The graph in Figure 2A 
shows some typical sound levels in a 1 -cycle band 
at 1 kc. 



KNOTS 


Figure 2A. Sound output of various ship targets. 


SONAR SEARCH-GENERAL 


77 


ECHO RANGING LISTENING 

2. Sound transmission: 

These sounds suffer a considerable loss in intensity in transmission through the water, a two-way trans- 
mission in the case of echo ranging, one-way for listening. During a passage from ship to target or vice versa, 
a certain loss is suffered due to spreading, attenuation, and refraction. Calculation of the transmission loss 
in any particular situation is a complicated problem. In the absence of refraction or reflection the intensity 
in decibels, I, can be expressed as a function of range, 

I(r) = 7(1) — 20 logior — ar. (1) 

The term 7(1) in equation (1) is the intensity at a distance of 1 yard from the source, 20 logior is the loss of 
intensity due to geometrical spreading of the sound (inverse square law), and ar is the loss due to absorption 
of energy by the water, a is the attenuation in decibels per yard (assuming r to be expressed in yards). This 
equation is strictly valid only if there is no reflection or refraction of the sound beam. In many cases, however, 
the effect of refraction can be represented by an increase in the value of a: assigning an ‘‘effective attenuation 
constant,’’ a may then be regarded as an empirical constant which depends on the frequency of the sound 
and temperature distribution in the ocean. The graph in Figure 2B gives values of the transmission loss, 
as calculated by equation (1) for some typical values of a. The effect of reflections from surface and bottom 
is neglected. A complete analysis of sound transmission would take into account reflection from various 
types of surface and bottom, and also refraction by temperature gradients whose effect cannot be repre- 
sented by an effective attenuation constant. Such an analysis is outside the scope of this chapter, but can 
be found in Volumes 6A, 7, and 8 of Division 6. 



100 1000 iqpoo 

RANGE IN YARDS 


Figure 2B. Transmission loss from 1 yard. 


78 


SONAR DETECTION 


ECHO RANGING LISTENING 

3. Reflecting power of the target: 

When the sound beam strikes the target, the 
amount reflected depends on the size and shape of 
the target, the nature of the target material, and 
also its orientation. The frequency and ping length 
of the sound being reflected are also of importance. 

These various factors determine the ‘‘target strength’’ 
which gives the intensity of the echo (reduced to 1 
yard from the target) relative to the intensity of the 
outgoing ping when it hits the target. For a sub- 
marine, typical values would be 

Bow aspect 16 db 

Beam aspect 25 db 

Stern aspect 8 db 

4. Receiver characteristics: 

The factors above determine the signal which arrives at the receiver. The characteristics of the gear, how- 
ever, have a great deal to do with the nature of the signal that is presented to the sonar operator. In general 
the gear will have a sensitivity that depends on frequency and the direction from which the sound is ap- 
proaching. At any frequency the directionality of the gear is determined by the physical properties of the 
receiving microphone (transducer) and its mounting. Both the transducer and selective elements in the 
receiver circuits come into play in determining the frequency response of the receiver. 

Factors Influencing Background Level 

1. Ambient noise in the ocean: 

Like the air which we inhabit, the ocean is not normally completely quiet, but full of various noises, man- 
made and natural. Chief among the man-made noises is that due to the searching craft, which will be dis- 



200 1000 2000 10,000 20,000 30,000 40,000 

FREQUENCY IN CYCLES 

Figure 3. Ambient noise. 


SONAR SEARCH-GENERAL 


79 


ECHO RANGING LISTENING 

cussed later. The true ambient noise due to natural causes is also considerable, being generated primarily by 
wave action. The average frequency distribution of this noise for various sea states is shown in Figure 3. 

Marine animals may also contribute to the ambient background. Famous for such activities is the snapping 
shrimp, a bed of which may produce a level of about 40 decibels above 0.0002 dyne per square centimeter 
in a 1 -cycle band above 5 kc. 

2. Self-noise created by searching craft : 

Since the receiving transducer is necessarily in close proximity to the searching craft, any noise generated 
by it will be heard as a contribution to background level. This self-noise may be caused by the propellers, 
by moving machinery in the ship, or by the rush of water about the face of transducer or dome. In any case, 
the self-noise increases rapidly with increasing speed (see Figure 4). At low speeds, ambient water noise 
often overrides it, but at high ship speeds, self-noise is the important factor. 



0 10 20 

SPEED IN KNOTS 

Figure 4. Background level for typical DD with dome. 



80 


SONAR DETECTION 


ECHO RANGING LISTENING 

3. Reverberation: 

The signal which is sent out by echo-ranging gear 
is reflected by many objects in the ocean besides the 
desired target. Irregularities of surface and bottom 
are the most important reflectors, but there are slight 
echoes from the body of water itself. The totality of 
these many false echoes is called reverberation. When 
reverberation is severe it may override other types 
of background noise and be the chief factor limiting 
the sonar range. Bottom reverberation from a rough 
or rocky bottom is the source of highest reverbera- 
tion level. Surface reverberation may also be con- 
siderable when the surface is rough and bubbly. Other 
factors such as ping length, frequency, and modula- 
tion also influence the reverberation level, as does the 
type of sound transmission to be found in the ocean 
at the time. 


4. Characteristics of sonar: 

The background level presented to the sonar operator depends on the sensitivity of the sonar, its frequency 
selectivity, and directionality. Since the sources of background noise do not in general have the same fre- 
quency distribution or the same bearing as the target, the frequency response and directionality of the gear 
affect background noise and signal differently; this may either facilitate or hinder target recognition. By 
choosing the frequency response so that the signal from the target is at the frequency of maximum sensitivity 
and training a directional type of gear so that its direction of maximum sensitivity is oriented toward the 
target the range of detection can be considerably increased. 


Factors Influencing Recognition of Signal 

For any given signal strength and background heard by the operator, there is a certain probability that 
he will recognize the signal. A very loud signal will surely be detected, a weak one has only a small chance. 
The level of signal relative to background for which the probability is 50 per cent is called the recognition 
differential. There are various means by which the signal is presented to the operator. In aural recognition 
the sound is presented to his ear by phones or a loudspeaker. Sometimes visual schemes are involved in 
which the signal is displayed as a spot on an oscilloscope or on sensitized paper. In any case the chief factors 
influencing it are the following: 

1. Type of signal: 

Length of ping, for example, is important in aural 
detection. A very short ping is not recognized as 
readily by the ear as a longer one. For visual detec- 
tion, as on an oscilloscope, this is no longer the case. 

Doppler frequency shift is valuable in differentiating 
aurally between signal and reverberation but is not 
useful with visual data presentation. Both the type 
of signal and the way it is presented to the operator 
are of importance. 


1 . Type of signal : 

The signal to be detected by listening gear varies 
widely in character from one target to the next. In 
broad-band listening, characteristic sounds such as 
propeller beats, gear whine, and machinery noises 
can be recognized even when their level is much lower 
than the background, and the recognition differential 
depends on the extent to which such noises aid in 
recognition. If the gear used for listening is sensitive 
only to a narrow band of frequency, however, the 
signal must be approximately equal to the back- 
ground to be recognized. 




SONAR SEARCH-GENERAL 


81 


ECHO RANGING 


LISTENING 


2. Type of background: 

The type of background may often influence the 
recognition differential. For instance, shrimp crackle, 
which is mostly high frequency, would not be very 
effective at masking low-frequency machinery sounds, 
and listening to them would have an unusually 
favorable differential with a background of that type. 


4. Operator skill: 

In the recognition of typical sounds from the target 
the skill of the listener plays a large part. He must know 
what he is listening for, a knowledge acquired only by 
training. Hence the recognition differential depends a 
great deal on the state of training and skill of the oper- 
ator. Fatigue and inattention can undoubtedly mate- 
rially change the differential. 

This enumeration of factors having to do with the effectiveness of sonar search gear is by no means com- 
plete. It is presented merely as an indication of the type of factor which should ideally be taken into account. 
If all these factors (and any others that are not mentioned) were accurately known at any instant, then it 
might be possible to decide precisely whether a target in a given position would be detected. As pointed out 
in Chapter 2, however, detection is never certain because of the human factor involved. We cannot tell 
whether detection will occur in a particular instance without knowing the detailed processes going on in the 
brain of the sound operator, and in addition we would have to be able to predict the acoustic behavior of 
the waves ahead of time, and to have already made extensive acoustic measurements on every target (usually 
an enemy craft) that might be encountered and to know when searching for it just which enemy craft 
was being sought. In any practical situation we can only estimate the probability that the target would be 
detected on the basis of the average values of various factors and their expected variation. 

Methods for expressing the probability of detection in quantitative terms are given in Chapter 2. They 
involve the use of an instantaneous detection probability coefficient y to take care of the human variable 
and the “short-term’’ fluctuations, viz., factors whose changes take place in a time that is short compared 
with the time taken by the full search operation. Normally there are also slowly varying factors that may, for 
instance, vary from day to day but are constant during the time of the operation. If these alone are of pre- 
dominant importance, so that the psychological and short-term effects can be ignored, it is convenient to 
assume that there is, in effect, a definite range at any time, and it is the distribution of these individual 
definite ranges which gives the overall lateral range curve. This is in accordance with Section 2.9 of Chapter 2. 

In order to show how some of these factors enter into the picture, two examples will now be discussed in 
some detail. The first of these is the expendable radio sono buoy [ERSB], a nondirectional listening device, 
the second is the standard directional echo-ranging gear used by antisubmarine ships. Similar treatments 
could be made and have been made for many other types of gear. 


2. Type of background: 

As has been indicated, the recognition differential 
is different according as the background is reverbera- 
tion or water noise. The doppler shift aids in dis- 
criminating against reverberation, since the ear can 
detect the difference in pitch. Water noise, however, 
contains all frequencies, so that doppler is of no help. 
Typical requirements for recognition differential are: 

a. With 0.1 — second ping vs background in 1-kc 
band: — 7 db; 

b. Vs reverberation, no doppler: -f-3 db; 

c. Vs reverberation 50 cycles doppler: — 4 db. 

3. Data presentation: 

Gear in which the operator actually hears the 
signal normally have different recognition differen- 
tials from those in which a spot on an oscilloscope 
or on chemical paper is the means of detection. 

4. Operator skill : 

Aural acuity and pitch perception enter into the 
question of recognition differential. Training in dop- 
pler recognition is of importance. Operator attentive- 
ness and fatigue are undoubtedly significant in rou- 
tine search operations. 


82 


SONAR DETECTION 


62 EXPENDABLE RADIO SONO BUOY 

The sonobuoy is a nondirectional sonic listening de- 
vice which is normally dropped from aircraft, floats 
at rest on the surface, transmits the sounds heard 
in the water to the plane via a radio link. Its use is to 
enable the aircraft to obtain sound contact with a 
submerged submarine. Because of its simplicity, it 
is suitable as a first example to illustrate the problems 
of sonar search, though it is not an altogether typical 
example of sonar search gear. 

There are two possible approaches to the problem 
of determining the lateral range curve for this, or 
any, form of detection. Either the curve can be cal- 
culated on an a priori basis from estimates of the 
factors involved, or it can be determined from op- 
erational data. The latter is more reliable when 
sufficient data have been gathered, but it is not always 
possible to do this. Calculations of the first sort are 
always valuable in that they throw considerable light 
on the importance of the various factors involved and 
help in the interpretation of operational results. In 
the following discussion a lateral range curve will be 
obtained on an a priori basis, and will then be com- 
pared with available operational data. 

The factors referred to in the previous section de- 
termine the chance of detection in any particular 
case. It is therefore necessary to assign values to 
them. The quantities required are: 

Pi = sound pressure of source at distance 1 yard 
in the sonobuoy band (0.1 to 10 kc); 

ju = transmission loss in traveling from source to 
sonobuoy; according to equation (1) this 
transmission loss [/(I) — /(r)] is given by 
20 logic r -h ar; 

Bl = background level received by sonobuoy, 
both water noise and any self-noise or cir- 
cuit noise in the buoy itself (in same band 
as used for Pi) ; 

= recognition differential, i.e., required signal 
level with respect to background for detec- 
tion to occur. 

The target will just be detected (i.e., detected with a 
50 per cent probability), if (values in decibels): 

Pi ~ M = Pl + 

( 2 ) 

or M = Pi - Pl - Az,. 

If all these quantities were definitely known, this 


equation would specify precisely a range within which 
the chance of detection exceeds 50 per cent, being 
considerably greater for most of this range. Approxi- 
mately, then, it would define a definite maximum 
range. Each of the quantities has a considerable 
range of variation, however. For instance, back- 
ground level is quite different for a sea of force 1 from 
what it is with force 5. Hence the expected sonobuoy 
range depends on sea state. An overall lateral range 
curve must give overall results, however, for those 
sea states met in practice. Each of the other factors 
will be variable also. The chief causes of variation in 
them are 

Pi actual speed of submarine is unknown, and 
sound output at a given speed varies widely 
from one submarine to the next; 

Bl background varies with sea state; 

A X, differential depends on the skill of the opera- 
tor and the type of noise made by sub; 

iu(r) depends on sound conditions. 

While these factors may change widely from one 
case to the next, it is evident that in any particular 
case they are more or less fixed. There is no major 
cause of variation which would be expected to give 
large changes during the time that the submarine 
is near the sonobuoy. (The opposite case is true for 
echo ranging, when the echo level may fluctuate 
broadly between successive pings.) For sonobuoys, 
however, we here neglect the small short-term fluctu- 
ations and the effect of “human fallibility,” and 
treat each combination of the variables as defining a 
definite range. On this basis we proceed to calculate 
the range for each combination of values for the 
important factors and obtain a distribution of these 
ranges corresponding to the assumed distribution for 
these values. This can be done conveniently if the 
values of Pi, Px,, and Ax, are assumed to be normally 
distributed. From available data, it is reasonable to 
assume values as follows : 




Mean 

Standard 

deviation 

Pi 

(decibels above 0.0002 dyne per centi- 
meter at 1 yd in sonobuoy band) 

128 

5 

Bl 

(decibels above 0.0002 dyne per centi- 
meter in sonobuoy) 

76 

4 

Al 

decibels 

-8 

4 


There is also considerable possible variation in /x(r). 
This can be represented by using a spread of at- 
tenuations, taking the attenuation equally likely to 


EXPENDABLE RADIO SONO BUOY 


83 


be anywhere between 0 and 3 db per kiloyard. We 
can combine the values of Pi, Bl, and to give 

Pi - Pl - Ax, = 60 ± 8 db. (3) 

Equation (3) can be solved graphically by a diagram 
of the sort shown in Figure 5. 

The. horizontal lines are drawn so as to divide the 
vertical axis into regions of 10 per cent probability, 


between 1 and 4 knots and individual variations of 
about 4 db in sound output of individual submarines. 
Bl corresponds to a sea state of 23 ^ with two-thirds 
of the cases falling between state 13^ and 33^^. As far 
as the overall result goes, however, it does not matter 
just what the source of variation is so long as the net 
result is a 0 - of 8 db. In order to show the effect of 
changes in sea state, for example, similar calculations 



100 500 1000 10,000 


RANGE IN YARDS 

Figure 5. Graphical solution of equation 3. 


and the transmission curves have the same property, 
that in any particular case there is one chance in 
three of the desired point lying in either of the three 
regions. These lines intersect forming cells, each of 
probability 1/30. If we assign to each cell the range 
of its midpoint, we have a theoretical distribution of 
ranges, and correspondingly a lateral range curve. 

The curves in Figures 6 and 7 represent expected 
overall results for a considerable range of submarine 
speeds, sea states, and other factors. The value of 
Pi, for instance, corresponds to a speed of about 3 
knots, with about two-thirds of the cases falling 


would be made for a series of fixed values of the sea 
state. This can be done for any factor whose in- 
fluence it is desired to study closely. 

We will, however, pass to the second, or opera- 
tional, aspect of the problem without going into 
further detail. The theoretical calculations are, in 
actual fact, not purely theoretical, but are based 
largely on tests of the gear and experimental runs 
under controlled conditions. Such tests must, of 
course, be made to give an idea of the operational 
performance of the gear and a firm basis for the- 
oretical predictions of its effectiveness. It is assumed 


84 


SONAR DETECTION 


that theory and test data will always be reconciled 
and put in agreement. The final check on the per- 
formance of the gear lies in the results of actual op- 
erations. This aspect of the problem will now be 
considered. 

Data on operational sonobuoy ranges cannot be 
obtained directly. If the report of the sonobuoy con- 
tact gives the buoy pattern, the period of time that 
the submarine was heard on one buoy or several 
buoys, and evidence, such as propeller beats, of 
submarine’s speed, then the detection range of the 
sonobuoy can be estimated. This has been done on 
the basis of data reported in connection with air- 
craft attacks on German U-boats, and on the basis 
of these estimates the following distribution of con- 
tact ranges was obtained. 


Range in Yards 
0- 499 
500- 999 
1000-1499 
1500-1999 
2000-2499 
2500-2999 
3000-3499 
3500-3999 
4000 and over 


Number of Cases 
0 
4 

4 
9 
9 
2 

5 
3 
3 


This is the actual distribution of ranges at which 
contacts were made, and is consequently prejudiced 
in favor of the longer ranges. If detection ranges of 
500 to 5,000 yards were all equally likely, many more 
5,000-yard contacts would be made than 500 — about 
ten times as many. Consequently, the distribution of 
potential contact ranges is obtained by dividing each 
of the above figures by the range. This curve of po- 



RAN6E IN YARDS 

Figure 6. Distribution of ranges (a priori). 



LATERAL RANGE IN YARDS 

Figure 7. Lateral range curve. 



STANDARD ECHO-RANGING GEAR 


85 


tential contact ranges corresponds to the theoretical 
curve for distribution of ranges, and is plotted below 
in Figure 8. A lateral range curve is also shown in 
Figure 9. It is evident that the operational ranges 
are, on the whole, better than those predicted on a 
theoretical basis. There are a number of possible ex- 
planations of this discrepancy. 

The most obvious explanation is to assume that 
some of our estimates of factors such as U-boat sound 


range contacts were lost than was assumed in ad- 
justing the distribution of actual contacts to obtain 
the “potential” range distribution. This would be 
the case, for instance, if contacts were lost because 
the submarine got past the sonobuoy while the ob- 
server was listening to a different one. Since it takes 
about 10 minutes, on the average, for a two-knot 
U /B to pass through the detection circle if the range 
is 500 yards, the chance of missing a U/B that does 



0 1000 2000 3000 4000 


RANGE IN YARDS 


Figure 8. Operational range distribution (adjusted). 



LATERAL RANGE IN YARDS 

Figure 9. Lateral range curves. 


output have been too conservative. An increase of 
four or five decibels in assumed mean U-boat sound 
output would account for most of the observed dif- 
ference. Similarly actual sea states may have aver- 
aged somewhat lower than was assumed, or the recog- 
nition differential may have been more favorable. It 
is not possible to decide just what the cause is with- 
out more detailed analysis or further information. 

It may be, however, that the reason for the dis- 
crepancy lies in our interpretation of the operational 
data. Perhaps a larger number of potential short- 


pass through the circle should be small. On the whole 
it seems most reasonable to conclude that operational 
results indicate that these theoretical values are 
somewhat too pessimistic, and they were, indeed, 
intended to be rather conservative. 

63 STANDARD ECHO-RANGING GEAR 

Echo-ranging gear such as is used by antisubma- 
rine ships is characterized by a slow intrinsic search 
rate due to its directionality and the slow speed of 


L 


86 


SONAR DETECTION 


sound (compared with electromagnetic waves). Only 
a small segment of ocean is searched by the gear in 
any short time interval (say one second) (see Figure 
10). Consequently there are only a small number of 
glimpses at any particular target: the point of view 
of glimpses, as outlined in Chapter 2, Section 2.2 
must be maintained. 

As was the case for sonobuoys, both a priori cal- 
culations and operational results must be used. More 
factors enter into the former, however, and the 
process outlined in Chapter 2, Section 2.9 must be 
applied. First, we deal with a situation in which the 
variable physical factors are fixed, i.e., the sound 



conditions, type of sound gear, orientation of sound 
beam, etc., are unchanging. There are, however, cer- 
tain factors which are bound to fluctuate, in par- 
ticular the intensity of echo on a particular ping, the 
intensity of background, and the operator’s ability to 
discriminate between echo and background. Even for 
given values of the former conditions, these latter 


fluctuations impart an uncertainty to the detection; 
and it becomes necessary to consider the probability 
of detection for the particular ping in question, the 
“one-ping probability.” Second, these values for the 
“one-ping probability” must be combined in ac- 
cordance with Section 2.4 (in the existing state of the 
science, the formal calculations being, of necessity, 
replaced by graphical methods) to give the prob- 
ability of detection by the particular search in ques- 
tion, a “fixed conditions probability.” In this way 
we can get, for example, a “fixed conditions lateral 
range curve.” By doing this we assume that the 
fixed conditions are indeed constant throughout the 
search and the fluctuating conditions do, in fact, 
fluctuate from ping to ping, so that the values for 
each ping may be chosen at random and indepen- 
dently from a suitable distribution. Third, the “fixed 
conditions probabilities” which result must be aver- 
aged over appropriate distributions of values for the 
conditions to correspond to our knowledge of them. 
If, for example, we desire to plot a lateral range 
curve for a particular ship, certain ship speed, bathy- 
thermograph pattern, sea state, but uncertain U-boat 
depth, speed, and operator alertness, we would pick 
the proper values of the certain factors, average over 
the assumed distributions of the uncertain ones, and 
obtain a curve. An overall average curve for all ships 
and conditions would require much more extensive av- 
eraging, and surely would appear to be quite different. 

We will now proceed to go through these steps for 
a typical example ; first the one-ping probability. 

In this one-ping probability all conditions are re- 
garded as fixed except for fluctuations in echo, back- 



WITH RESPECT TO IDEAL RECOGNITION LEVEL 

Figure 11 . Dependence of probability of detection of one ping on the sound level above background. 


STANDARD ECHO-RANGING GEAR 


87 


ground, and operator alertness. The effect of these 
fluctuations can be represented by a curve (Figure 
11) which gives the probability of recognizing the 
fluctuating echo as a function of its average level. 
This average level is given on a scale relative to 
the background. The zero on the scale is the echo 
level which would ideally be required for recognition 
if no fluctuations occurred. It is now necessary to 
calculate the echo level with respect to background 
as a function of target position in the sound beam. 

The target’s position in the sound beam may be 
specified by two variables, range and relative bearing. 
Range introduces a transmission loss as in Figure 2B, 
and bearing a loss relative to the axis because of 
directionality. As in Figure 5, various factors must be 
taken into account to determine the admissible loss 
which will just give recognition (i.e., with a 50 per 
cent chance), fluctuation neglected. These factors are 

Pi transmitted signal strength at 1 yard; 

Bp background level heard in gear if reverbera- 
tion is limiting; Rp is higher for short-range 
echoes than long, i.e., a function of range; 

T average target strength; 

Ap ideal recognition differential for nonfluctu- 

ating case; 

/x(r,i3) transmission loss relative to unit range on 
the axis (see Section 6.2). 


Pi — fj, T = Bp + Ap (4) 

= Pi T — Bp — Ap. 

It is necessary to assume values for these factors 
in accordance with the specific case in question. For 
simplicity, we will assume that the transmission loss 
can be represented by inverse square law with 7 db 
per kiloyard attenuation, as might be the case for a 
target below the thermocline. Using this assumption 
and the typical directivity pattern of Figure 12, 
we can draw in the curves for ju as a function of r for 
a number of values of jS. 

The solid curves of Figure 13 are drawn on this 
basis. Values of (Pi + T — Rp — Ap) must also be 
plotted to give a graphical solution of equation (4). 
Taking into account the probability of detection 
curve of Figure 11 we can add or subtract suitable 
values from (Pi + T' — Rp — Ap)to give the trans- 
mission loss for 0 per cent, 20 per cent, 40 per cent, 

• • • probability of contact. Intersections of these 
dotted levels with the transmission loss curves give 
the value of the probability of detection, i.e., the 
one-ping probability, for the various ranges and bear- 
ings. 

On the one-ping probability curves (Figure 14), all 
curves have been terminated at a minimum range 
of 500 yards. Since the submarine is thought to be 
deep, the likelihood of contact at ranges shorter than 


330“ 0 30“ 



Figure 12» Directivity for typical projector. 


88 


SONAR DETECTION 


500 yards is not great. There is a small area in which 
contact is theoretically certain, but the greater part 
of the beam has probabilities of 0 per cent to 80 
per cent. The assumptions upon which this one-ping 
probabihty function is based are the following: 

Pi 180 db, corresponding very closely to the 
signal output of standard gear. 

Bp 40 db in a 1-kc band, heard in sound gear. 
This takes into account the effect of direc- 
tionality in discriminating against noise back- 
ground; it corresponds to typical self-noise 
at a speed of about 15 knots. No reverbera- 
tion is considered. 


T 15 db, a value which is typical of a submarine 
except at beam incidence. 

Ap — 7 db, the value obtained in laboratory tests 
for recognition differential relative to noise in 
a 1-kc band. 

Having obtained the one-ping probability for the 
conditions under consideration, we must now obtain 
the corresponding lateral range curve. The method 
of Chapter 2 applies in principle, but cannot readily 
be carried out exactly in terms of formulas in the 
present case; it is necessary to resort to graphical 
methods of calculation. Suppose that the one-ping 
probability function is represented by a group of 



RANGE IN YARDS 

Figure 13. Graphical probability calculation. 


STANDARD ECHO-RANGING GEAR 


89 


270* 300* ‘ 330* 




RANGE IN YARDS 

Figure 15. One-ping probability diagram. 



90 


SONAR DETECTION 


dots, the density of dots being proportional to the 
probability. For example, in the diagram (Figure 15), 
10 per cent of the squares between 0 per cent and 
20 per cent contours have dots in them, 30 per cent of 
those between 20 per cent and 40 per cent, and so on. 

We now decide on a planned sequence of pings, 
i.e., starting with projector trained abeam and train- 
ing forward 5 degrees between pings until the bow is 


range, we can obtain the lateral range curve. The 
result for this case is shown in Figure 17 at the same 
scale as previous lateral range curves. This accord- 
ingly completes the second step in the required cal- 
culations. 

Before discussing the third phase of the echo- 
ranging gear picture, it is worth while to point out 
the practical applications of the ‘‘fixed conditions 


jtn 

I 

</) 


LATERAL RANGE IN YARDS 


m 


3000 



Figure 16. Coverage diagram (for one cycle of plan). 


reached, and then slewing aft to the beam on the 
other side and pinging forward again by 5 degree 
steps. If we assume that the one-ping probability 
is not changed by training the projector from one 
bearing to the next, and assumed a relative velocity 
for searcher and target, then we can lay out suc- 
cessive pings in target space (i.e., space fixed rela- 
tively to the target). Target space is divided into 50 
yard squares, and the event of a dot on the one-ping 
probability diagram filling in a square means that a 
submarine in that square would be detected. This 
process can be followed mechanically in the follow- 
ing fashion. Transparent paper divided into 50-yard 
squares is placed over the one-ping probability dia- 
gram in position for the first ping. Each square in- 
cluding a dot is blacked out. The diagram is moved 
to position for the second ping, and the process re- 
peated for all pings. This gives an area (see Figure 
16) black near the center and white at long range, 
the fraction of squares blacked out giving the prob- 
ability of catch at any region. Such a diagram will be 
periodic if the pinging plan is regular, and by averag- 
ing the probability over one period for a given lateral 


lateral range curve” calculations. In any tactical 
situation some of the conditions are reasonably well 
known, ship speed, sea state, bathythermograph 



0 1000 2000 3000 


LATERAL RANGE IN YARDS 

Figure 17. Lateral range curve (calculated from one- 
ping probability). 

record, and so on. There are other conditions which 
can be varied by the searching craft, namely the 
plan of search, i.e., keying interval used, limits of 
sweep, number of degrees to turn between successive 




STANDARD ECHO-RANGING GEAR 


91 


pings, spacing between ships. By calculating lateral 
range curves for typical values of the tactical condi- 
tions the best values of these controllable variables 
can be determined for the various tactical situations. 
In this way it can be shown, for instance, that the 
sweep should not normally be restricted to a small 
angle either side of the bow, but should be started 
at least as far aft as the beam, and should always 


tire sweep from beam to bow as a single ‘‘glimpse’^ 
with a fixed probability of detection. 

When, however, operational results are to be com- 
pared with theoretical predictions, it must be kept 
in mind that the conditions are by no means fixed. 
This distinction was discussed in Section 2.9. There 
is a great variety in the actual conditions — the sonar 
involved, the sound conditions, and submarine’s 



0 1000 2000 3000 4000 


LATERAL RANGE IN YARDS 

Figure 18. Operational lateral ranges for echo-ranging gear. 


be made from aft forward rather than in the opposite 
direction. Many calculations of this sort have been 
made to determine the proper tactical use of echo- 
ranging gear in search. Detailed results of these cal- 
culations are not of interest at present — the general 
principle involved is that the gear should be used so 
as to give uniform coverage of the area without de- 
veloping serious gaps, as must obviously be the case 
from Chapter 3. 

In many cases calculations have been made using 
rather rough approximations. The one-ping prob- 
ability function may be replaced by a simpler one 
which is zero outside a certain contour and equal to 
a constant (usually about 0.5) inside. This gives a 
more abrupt lateral range curve than the more ac- 
curate calculations, but most conclusions concerning 
best operation of the gear are not changed. An even 
simpler approximation involves considering the en- 


depth, speed, and reflecting power. In principle, it 
is necessary to calculate lateral range curves for all 
values of the many variable conditions, and then 
average them appropriately. The labor involved in 
such a calculation would, however, be completely 
prohibitive. In addition, it would be necessary to 
know what distributions of all the important vari- 
ables are met with in operations, and this informa- 
tion is not available. Consequently we will only pre- 
sent the operational results for comparison with 
Figure 17, and then determine what range of condi- 
tions might give the observed results, and whether 
this range does, indeed, appear reasonable. 

As a fair sample of operational data, 235 first con- 
tacts by echo ranging in the period 1 January 1943 
to 30 June 1944 can be quoted. From these contacts 
the number of contacts versus lateral range is plotted 
in Figure 18. 


92 


SONAR DETECTION 


This curve indicates considerably shorter ranges 
than that of our previous example which led to the 
lateral range curve of Figure 17. In other words, the 
conditions which led to the one-ping probability 
curve of Figure 14 were more favorable than most of 
those met with in practice. The operational curve of 
lateral ranges is, however, of the general type that 
would result from averaging a number of lateral 


ized to give the same probability at 1,000 yards. As 
can be seen, this calculated curve is in excellent 
agreement with the observed data. The agreement is 
largely fortuitous and certainly does not imply any 
essential correctness of the calculated curves. It does, 
nevertheless, indicate that the operational results are 
in keeping with the physical picture if, and only if, it 
is assumed that short and medium sonar ranges pre- 



range curves like that obtained for fixed conditions. 
Figure 19 above shows a number of lateral range 
curves of the same general type as Figure 17 which 
have 50 per cent ranges of 500 yards, 1,500 yards, 
and 2,500 yards. A combination of these curves in 
the proportion 5:3:1 leads to a lateral range curve 
of very much the same shape as that obtained from 
operatioi^al results. In Figure 20 this theoretical 
curve is compared with an operational curve normal- 



Figure 20. Lateral range curves, overall average. 


dominate in actual operations. The sweep width ob- 
tained from these curves is 1,800 yards, which is 
smaller than that usually obtained from purely 
theoretical considerations. This no doubt reflects 
frequent unfavorable sound conditions, imperfect 
maintenance of gear, reduced operator skill, and 
similar factors more or less inevitable under operating 
conditions. 

64 PARALLEL SWEEPS 

When a number of searching units are available, 
they normally operate together so that their paths 
in target space form parallel sweeps. For example, a 
line of sonobuoys which the submarine crosses ef- 
fectively carry out parallel sweeps relatively to the 
submarine; so do a group of ships sweeping in line 
abreast. For such parallel sweeps, the lateral range 
curves of the individual units must be combined to 
give an overall probability of detection curve. The 
manner of doing this depends on the physical situa- 
tion — whether the lateral range curve arises from 
variable or fluctuating conditions, or both. This point 
has been discussed in Section 2.9. 


PARALLEL SWEEPS 


93 


Consider two units making parallel sweeps, spaced 
S yards, with a submarine penetrating, somewhere 
between them at point x. Then the probability of 
detecting the submarine is given by 

P{x) = p(x) + p{S — x) — p{x) p{S — x) (5) 

if the probabilities of sighting are independent. In 
some cases, however, the probabilities are by no 


ing ships; consequently the probabilities are com- 
bined in accordance with equation (5). In actual fact, 
conditions are rarely truly ‘ ‘fixed’ ^ because the depth 
of submarine, for instance, is not usually known. 
Nevertheless, calculations made for fixed conditions 
of typical values are useful in deciding on proper 
ship spacing. In doing this, the assured range is 
normally employed. This is defined as follows: 



means independent. In the case of two sonobuoys, 
for instance, the lateral range curve depends on the 
distribution of submarine sound outputs. A sub- 
marine that is noisy for sonobuoy at A is also noisy 
for a sonobuoy at B (see Figure 21). To the extent 
that variable conditions are in each case the same 
for both searchers, the two probabilities are alto- 
gether dependent. In this case the combined prob- 
abihty at x is simply p{x) or p{S — x), whichever is 
larger. 

For echo-ranging search under fixed general condi- 
tions, the lateral range curve arises from rapid fluctu- 
ations in the residual uncontrollable conditions, thus 
resulting in independent probabihties for the search- 


Consider the maximum range obtained by standard 
range prediction methods at a given depth ; then take 
the minimum such range as the depth is varied, 
passing through possible depths, i.e., the maximum 
range at the most unfavorable depth : this is defined 
as the assured range. Accordingly, ship spacings 
based on this range will be rather tight and con- 
servative. The lateral range curve of Figure 17 cor- 
responds to an assured range of 2,000 yards. The 
derived curves in Figure 22 show the combined prob- 
ability function for two ships with various ship spac- 
ings. 

These curves can be interpreted in a number of 
ways. If the submarine can always choose the best 


PROBABILITY 
OF DETECTION 



Figure 22. Probability versus lateral range for a pair of ships with various spacings. 




94 


SONAR DETECTION 


pwnt to try to sneak through, then it is the minimum 
value of these curves that counts, i.e., measures the 
tightness of the screen. If, on the other hand, he 
passes between more or less at random, then it is 
the average value which is important. If the number 
of ships is small, the submarine can often evade them 
by steering to the side at high speed and passing 
around the end. This type of evasion is neutralized 
to some extent by placing a group of ships in line 
abreast, the resulting front being too broad for the 
submarine to end-run readily. Increased ship spacing 
gives a broader front at the expense of holes between 
the ships. 

From the curves in Figure 23 it is apparent that 
a ship spacing of 1 3 ^ times the assured range is very 


From a practical point of view another factor 
should be considered. The operational data indicate 
that rather short ranges predominate in actual opera- 
tions, and hence that the theoretically predicted ‘‘as- 
sured range” may be somewhat optimistic. On this 
basis the rather conservative ship spacing of 13 ^ to 
times the assured range may be altogether justi- 
fied. 

Evidently the choice of ship spacing for a screen 
will be different from that for a search (or hunt). The 
former is defensive, and its primary measure of 
effectiveness is its ability to intercept submarines 
which are attempting to penetrate to the proximity 
of the screened units. The latter is offensive, and its 
measure of effectiveness is the expected number of 



SHIP SPACING IN TERMS OF ASSURED RANGE 

Figure 23. Probability of detection as a function of ship spacing. 


tight, and spacings over 2 times have rather danger- 
ous holes. A complete analysis of the question of 
optimum ship spacing would involve the size of area 
to be searched, the submarine’s ability to end-run 
and choose weak points in the screen, and the quan- 
tities presented in the curves. As a rough rule of 
thumb, a spacing of 1^ times the assured sound 
range is now specified in doctrine for searches in line 
abreast, 13 ^ times (when possible) for screens. 


submarine contacts which it produces. In view of the 
overlapping effect of close spacing, this expected 
number will be reduced, whereas the tightness of 
the screen will be increased when spacing is close. 
Thus screens will normally employ tighter spacing 
than hunts. The end-run prevention cited above is a 
further argument for wider spacing in the case of 
hunts than in that of screens. This whole question 
will be entered into in great detail in Chapter 8. 


Chapter 7 

THE SEARCH FOR TARGETS IN TRANSIT 


The General Question 

I N PLANNING A SEARCH, the nature of the target is 
usually known, and its general position may be 
more or less known as a matter of probability (as in 
the problems of Chapter 3) ; but unless a fairly defi- 
nite estimate of its motion can be made, the plan of 
search will have to be designed so as to be effective 
against a target having any one of many different 
sorts of motion, rather than being particularly ef- 
fective against targets of some one special kind of 
motion and less so against others which are recog- 
nized as irrelevant to the tactical problem in hand. 
The emphasis of this chapter is on the latter situa- 
tion, which arises when both the intent and the cap- 
abilities of the target are known. To know the intent 
of the target is to know where it is going: through 
what part of the ocean it passes, from what geo- 
graphical locality it comes, to what place it is going, 
etc. And to know the target’s capabilities is to have 
a reasonably good estimate of its speed u, as well as 
its endurance, etc. An essential part of such informa- 
tion can be put mathematically as follows: The tar- 
get’s vector velocity u is known at the different parts 
of the ocean where it is expedient to conduct the 
search. And since the main objective is to prevent 
the target’s undetected accomplishment of its in- 
tention, success will be achieved even if the target 
is not detected but is forced to abandon its objective 
in order to avoid detection. 

Attention will be confined in this chapter to the 
case where the detectability of the target does not 
change with the time, at least during long periods; 
thus in the case of visual or radar detection, surface 
craft (including surfaced submarines) are alone con- 
sidered; while in the case of sonar detection, the 
submarine target is regarded as constantly sub- 
merged. This avoids the great complication which 
would occur, for example, in the case of a submarine 
whose tactics of submergence and emergence are not 
known and, since they may depend on the tactics of 
search, could only be evaluated by some form of 
“minimax” reasoning. Thus “gambits” are not con- 
sidered herein. 

Three cases are of great importance in naval war- 
fare and will be studied in the three parts of this 


chapter. In the first, the target’s intention is to 
traverse a fairly straight channel (which may be a 
wide portion of the ocean); the vector velocities at 
all points are parallel and equal (a “translational 
vector field”); the search is called a harrier patrol. 
In the second case, the target is proceeding from a 
known point of the ocean (e.g., a point of fix, an 
island, or a harbor); the vector velocities are equal 
in length but are all directed away from this point 
(a “centrifugal radial vector field”); to this class of 
search belongs the trapping square and the retiring 
search when the approximate time of departure is 
known, and closed harriers, etc., in other cases. In the 
third case, the target’s intention is to reach a definite 
point (e.g., the seat of a landing operation, an island 
needing supplies, a harbor) ; the vector velocities are 
equal in length but directed inward toward this point 
(a “centripetal radial vector field”) ; again the method 
of countering this intention maybe the closed barrier. 
There are of course various cases closely allied to 
the three just mentioned, such as the antisubmarine 
or antishipping hunts conducted by carrier aircraft 
as the carrier sweeps through the ocean. But when 
the principles of this chapter are understood, the de- 
sign and evaluation of such plans offer little difficulty. 

In the case where the target intends to reach a 
point moving on the ocean (a ship, convoy, or task 
force), the vector field is of an entirely different char- 
acter; the form of search used is then called a screen. 
It forms the subject of Chapters 8 and 9. 

Throughout the present chapter, the effect of tar- 
get aspect on detection is disregarded. In the case 
of radar detection, for example, the possibility may 
exist of securing a somewhat higher chance of de- 
tection of targets of certain restricted velocity classes 
by using specially selected tracks; but the greatly 
added complication does not appear to warrant their 
consideration here. 

7 1 BARRIER PATROLS 

^ ^ ^ Construction of the Crossover Patrol 

Under a wide variety of circumstances, the prob- 
lem of detecting targets in transit through a channel 
by means of an observer whose speed v considerably 


95 


96 


THE SEARCH FOR TARGETS IN TRANSIT 


exceeds the speed u of the target (e.g., an airborne 
observer and ship target) can be simplified to the 
following mathematical statement. Given a channel 
bounded by two parallel lines L miles apart (the 
fine vertical lines of Figure 1) and given targets mov- 
ing through this channel and parallel to it at the 
fixed speed u (downward in Figure 1); how shall ob- 
servers fly from one side of the channel to the other 
and back, etc., in order to be most effective in de- 
tecting the targets? It is usually necessary to attach 
the flights to a fixed reference point 0 from which 
they start or take their direction. Thus 0 may be 
a conveniently recognizable point at or near the 
narrowest part of the actual channel (which will in 
general correspond only approximately to the mathe- 
matically simphfied parallel channel shown in Figure 
1), 0 may be a harbor or air base, etc. It is convenient 

TARGET 



A A, » o’o; = w 
AB = C D s M 

Figure 1. The crossover barrier patrol designed to be 
tight. 

to draw the line 00' (dotted in Figure 1) across and 
perpendicular to the channel, a purely mathematical 
reference line called the harrier line. 

The reasoning leading to the construction of a 
barrier patrol was based, historically, on the definite 
range law of detection (see Section 2.2; search width 
W = 2R). Since it leads in a natural manner to a 
form of patrol (the crossover barrier patrol) which 
turns out to be the optimum form from the point of 
view of any not-too-asymmetrical law of detection, 
it will be followed in detail in this section, while its 
more reahstic evaluation will be considered in Sec- 
tion 7.1.2. 


For convenience of wording, the target will be 
referred to as a ship and the observer as an aircraft. 
While this corresponds to the most important case, 
others will be considered later. The same mathe- 
matical ideas apply in all cases. 

Consider those targets which at the initial epoch 
(t = 0) are on the barrier line 00'. An observer 
starting from 0 when t = 0 and wishing to fly over 
these targets will not succeed in doing so if he flies 
along 00', except in the excluded case of targets at 
rest {u = 0). He will have to fly along OA, where the 
angle O'OA = a, called the lead angle, is determined 
by the requirement that the observer reach each 
point on A B at the same time that the target which 
was initially on the point of 00' directly above it 
reaches that same point, i.e., sin a = u/v, 

• -1 ^ ( 1 ) 

a = sin ^ 

V 

When the observer flies along OA, he detects not 
only the targets initially on 00', but, in virtue of the 
definite range assumption, all those within a distance 
of IF/2 miles on either side of 00': the band of width 
IF centered on 00' (Figure 1) is swept clean; i.e., 
all the targets which may have been in this band 
when t = 0 are detected. 

The observer now wishes to detect, on his return 
flight, all those targets which were, when ^ = 0, in 
a second band of width IF adjoining the first one and 
directly above it. This band is centered on the line 
OiO'i of Figure 1, where OOi = O'O'i = IF. The ob- 
server reaches A when 

^ ^ ^ 

V V cos a \/ 

At this epoch, a target initially at O'l, IF miles above 
O', will have moved down to a point Ai, but continue 
to be IF miles above the point A which the target 
initially at 0' will have reached by this time; AAi = 
W. This is because if one target is IF miles behind 
another, and if they both have the same speed and 
course, it will always be IF miles behind. Obviously 
if the observer were to fly directly back to the left 
bank of the channel, he would not fly over the O'l 
target (now at Ai), and hence not accomplish his 
purpose of sweeping the OiO'i band of targets. To do 
this, he must fly up the right bank, until he meets 
this O'l target at a point denoted by B and de- 
termined by the condition that the time taken for 
the observer to fly from A to B equals the time taken 


BARRIER PATROLS 


97 


for the target to go from Ai to B, i.e., AB/v = AiB/u. 
This equation together with the fact that AB + 
AiB = W determines the length of upsweep M = AB; 
solving these equations we find 


M = 


V 

V u 


W. 


( 2 ) 


This flight takes M/v = W/{v + u) hours, so that 
the observer is at B when 


t = 


a/ 


+ 


W 

V + u 


At this epoch, the targets, which when ^ = 0 oc- 
cupied the OiO'i band, will be in a band of width W 
centered on the line (not shown in Figure 1) through 
B perpendicular to the channel. From then on the 
situation is precisely similar to what it was initially : 
the observer will sweep this band clean if he flies back 
to the point C of the left bank, where the lead angle 
of BC has the same value a as before. And having 
arrived at C, he must make the upsweep CD = M 
if he is to detect on his third crossing the targets 
which when t — 0 were in a third W width band 
adjacent to and above the OiO'i, i.e., the band cen- 
tered on 020'2, Figure 1. 

The time taken by the target to fly the basic 
element OABCD is denoted by To; the time computa- 
tions of the preceding paragraph show that 


r„= 2^ + ^. 

\/v^ — V u 


( 3 ) 


bisects OA and BC; the flights if continued by the 
same aircraft would repeat themselves exactly; the 
path OABCD = OABCO would be flown over and 
over again, and thus the barrier would remain sta- 
tionary; this is called the symmetric crossover barrier 
patrol, or, if one will, the stationary element barrier. 

Figure 2C. D is above 0; then Tt is greater than 
To, and the first crossover point X is to the left of 
the center of the channel or, in extreme cases, may 
not occur at all (the second, to the right, etc.); the 
flights if continued by the same aircraft would take 
place farther and farther up the channel and thus 
lead to an advancing element barrier. 

It is to be emphasized that all three barriers may 
be flown as stationary barriers by the device of re- 
peating the elements by having successive aircraft 




There is another interval of time which it is useful 
to consider: the time Tt which a target takes in 
moving from O 2 to 0. Since OO 2 = 2W we have 


Figure 2 illustrates the three possible cases. They 
are as follows: 

Figure 2 A. D is below 0; then Tt is less than To 
(since the O 2 target and the observer reach D simul- 
taneously), and the first crossover point X is to the 
right of the center of the channel (the second, to the 
left, etc.) ; the flights if continued by the same aircraft 
would take place farther and farther down the chan- 
nel and thus lead to a retreating element barrier. This 
is the case for which Figure 1 has been drawn. 

Figure 2B. D coincides with 0; then Tt = To, 
and the point X is at the center of the channel and 


B THE SYMMETRIC BARRIER 



Figure 2. The three cases of crossover barrier patrol. 

start from 0 at epochs of Tt after one another: 
While the elements themselves may retreat or ad- 
vance, the geographical position of the flights, and 
hence of the barrier as a whole, remains stationary. 
This will be illustrated by later examples. 

The advancing barrier represents a situation in 
which more than enough flying is available (assum- 


98 


THE SEARCH FOR TARGETS IN TRANSIT 


ing the single aircraft’s endurance sufficient for the 
repeated flights) to produce the required coverage. 
Advantage can be taken of this circumstance to fly 
only during the favorable hours of the twenty-four, 
in daylight, if the greatest chance of detection or 
certainty of recognition is the main desideratum, at 
night with radar, if the element of surprise is more 
important; etc. For after advancing the barrier 
sufficiently during the favorable period, flights can 
be discontinued and the unswept waters (target posi- 
tions) can be allowed to come down until their lower 
boundary reaches 00' (Figure 1), whereupon the 
flights are recommenced. To find the time of no pa- 
trolling, one can reason as follows: If one basic ele- 
ment OABCD is flown, but not a second, the central 
axis of the first unswept strip (the 020'2 strip of 
^ = 0) will require the further time Tt — To to reach 
00'; if N basic elements are flown, the further time 
N{Tt — To) to reach 0 . However, if the last upsweep 
is not flown, it will require M/v = W/(v -h u) more 
time, since the aircraft completes its patrol flights 
that much earlier. Hence the interval of time (after 
the aircraft reaches the left bank for the last time) 
during which no patrol flights (as distinguished from 
the return flight to base) need be flown is N(Tt — To) 
-h W l{v -f u). With the aid of (3) and (4) we obtain 
the expression. 

No patrolling period = 

2vW 2L 1 ^ W 

u{v u) V u 

The symmetric barrier represents a situation in 
which the continued flying of the single aircraft is 
exactly enough to ensure the required coverage. It 
has a great advantage of simplicity over the asym- 
metrical cases ; and the measures described later for 
making it applicable are frequently taken. 

The retreating barrier represents a situation in 
which a single aircraft, even if its endurance would 
permit it to fly a large number of basic elements, is 
insufficient to maintain the required coverage, since 
the unswept area invades positions farther and farther 
down the channel. Under these conditions its flight 
has to be supplemented by that of other aircraft. 
One obvious way of doing this is to have a second 
aircraft leave 0 at the time when the target initially 
at O 2 reaches 0 , i.e., when t = Tt = 2W ju. This will 
be To — Tf hours before the first aircraft reaches D, 
and still longer before it returns to the base 0 . The 
second aircraft flies OABC, etc. But a better plan is 


to have n aircraft fly simultaneously abreast in a 
line perpendicular to their course and at the distance 
W apart. This has the effect of increasing the search 
width to the value W' = nW, and thus, if n is suffi- 
ciently great, leads from a retreating element barrier 
to a symmetric or an advancing one. Let us assume 
that with one aircraft Tt < To, i.e., the barrier 
element is retreating. What is the least number n 
of aircraft flying as described which will give rise to 
a nonretreating one? The answer is found by im- 
posing the condition Tt ^ To and replacing To and 
Tt by their expressions in (3) and (4) in which W 
has been replaced by W' = nW. We have 

2nW 2L , 2nW 
^ - - H > 

U — V? V u 

which is transformed algebraically so as to give the 
condition 

( 6 ) 

W V \v — u 

Since n is an integer, it is taken as the lowest integer 
greater than or equal to the expression on the right, 
which is in general not an integer. Thus the number 
of aircraft is proportional to the width of channel 
and inversely proportional to the search width; and 
when V is so much greater than u that the radical 
can be regarded as unity, it is proportional to the 
target’s speed and inversely proportional to the ob- 
server’s speed. 

It is remarked that when the width of channel L 
is not overwhelmingly greater than the search width 
W, an attempt is sometimes made to base the cross- 
over patrol not on the boundary lines of the channel 
as in Figure 1 but on two lines parallel to them and 
at a distance W /2 to the right of the left-hand 
boundary and IF/2 to the left of the right-hand 
boundary, respectively. But all the discussion and 
formulas previously given apply to this case, pro- 
vided L is replaced throughout by L' = L — W. 
Thus (6) becomes 

\IF / V \v — u 

And we derive all the formulas needed to consider 
the altered channel case together with the case of n 
aircraft abreast, simply by replacing IF by nIF and 
L by L — IF in formulas (2) and (5). Thus, the 
length of upsweep formula (2) becomes 


BARRIER PATROLS 


99 


M 


V + U 


and the conditions for cases of Figures 2A, 2B, or 2C 
are that Tt < To, T = To, or T > To take on the 
forms 

nW ^ L - W , nW 

A — / = + ’ 

u -s/ — V? V u 

nW L - W ^ nW 

R — — / ~r > 

u -y/ — V? V u 

nW ^ L — W nW 
u 's/ V u 


We have seen in Figure 1 how the bands swept 
clean according to the definite range law exactly 
cover the channel without overlapping or holes. That 
a plan of parallel flights, which, being intuitively 
simpler, might be tried instead of crossover flights, 
produces overlapping and holes and thus a loss both 
of efficiency and effectiveness, is illustrated in Figures 
3A, 3B, 3C, 3D, and 3E. The geometric object 
lesson contained in these figures was of historical 
decisiveness in an important operation of World 
War II. Here the problem was to set up a barrier 
effective against a 24-hour run at about 12 knots 
of enemy blockade runners in a known direction. 


ENEMY'S 

COURSE 



Figure .3A. Ineffectiveness of parallel search courses as barrier (legs separated by twice detection range). 


100 


THE SEARCH FOR TARGETS IN TRANSIT 


ENEMY'S 

COURSE 



speed of plane 150 knots reliable detection range 35 miles 

speed of ship 11.7-12 knots line patrolled 600 miles 

plane separation 523^ miles (3 planes) 

Figure 3B. Ineffectiveness of parallel source courses as barrier (legs separated 1.5 times detection range). 


ENEMY'S 

COURSE 



speed of plane 150 knots reliable detection range 35 miles 

speed of ship 11.7-12 knots line patrolled 600 miles 

plane separation 523^ miles (3 planes) 

Figure 3C. Ineffectiveness of parallel search courses as barrier (direction of flight reversed). 


BARRIER PATROLS 


101 


ENEMY’S 

COURSE 



speed of plane 150 knots reliable detection range 35 miles 

speed of ship 11.7-12 knots line patrolled once daily 600 miles 

Figure 3D. Barrier patrol without holes. 


ENEMY'S 

COURSE 



speed of plane 150 knots reliable detection range 35 miles 

speed of ship 11.7-12 knots line patrolled once daily 600 miles 


Figure 3E. Preferred type of barrier patrol. 


102 


THE SEARCH FOR TARGETS IN TRANSIT 


^ Crossover Barriers with Any Law 
of Detection 

The assumption of a definite range law of detec- 
tion, while affording a convenient basis for the con- 
struction of barrier patrols, leads to one fallacious 
conclusion, namely, that if designed as in Section 7.1.1 
it is absolutely tight, providing a 100 per cent chance 
of detecting the target; and, on the other hand, if it 
is designed with a slight overestimation of the search 
width IF, it has holes. This in turn may lead to the 
practically disadvantageous procedure of exerting 
great effort to basing the barrier on a preconceived 
value of W, believing that nothing short of this 
tightness is adequate, while at the same time hav- 
ing a false sense of security when this ideal is 
achieved. The matter at issue is, in other words, just 
what it was in Chapter 2, when the importance of 
considering various more realistic laws of detection 
was stressed. The fact is that no barrier is 100 per 
cent tight, whereas one falling far short of the ideal 
of the cleanly swept adjacent strips of Section 7.1.1 
may have very real value; it may provide a very 
useful probability of detection. Thus if a barrier de- 
tects on the average even one-quarter of all targets, 
it will make it a very dangerous and costly procedure 
for the enemy to send his shipping through the 
channel. 

In order to apply the machinery of Chapter 2 to 
these barrier patrols, we shall consider how the flights 
of Section 7.1.1 appear in space relative to the targets, 
i.e., in a horizontal plane moving down the channel 
with the speed u, “ 2 ^-moving space,” as we shall say 
for brevity. In such a space, all the targets are fixed. 
Thus the lines of targets 00', OiO'i, 0^0' 2 , etc., are 
stationary horizontal lines, maintaining for all values 
of t their positions as shown in Figure 1 for ^ = 0. 
And since, as we have seen, an observer flying the 
basic element OABCD of the plan passes directly 
over the 00' and O'lOi targets in the order of these 
letters, as well as up segments of the boundaries of 
the channel in passing from the 0' target to the O'l 
target, and from the 0i target to the O 2 target, the 
basic element flight will be along the path 00 ' 0 'i 0 i 02 
of it-moving space, as shown in Figure 4. And as 
long as the crossover patrol is flown, more and more 
of the horizontal lines and their connecting segments 
of Figure 4 will be traversed. 

But this is simply the case of detection of a 
randomly placed stationary target by means of 
parallel sweeps, the problem considered in Section 


2.7. The fact that the distance between parallel 
tracks comes out as W in Figure 4 is merely a con- 
sequence of the assumption of the definite range law 
made in Section 7.1.1 above, along with the desire of 
making the barrier 100 per cent tight. The point to 
be emphasized is that the crossover harrier patrol gives 
the best distribution of flights — uniformly spaced par- 
allel sweeps; this is why, in spite of rejecting one part 
of the assumptions upon which its design was based 
(definite range law), the form of patrol is still re- 
garded as optimum. But in the interest of flexibility, 
to provide coverage by parallel sweeps when a sweep 
spacing of aS = IF cannot be used, one must get 
away from this particular value of S. 

Let the sweep spacing S he an arbitrarily chosen con- 
stant, one chosen without necessary relation to IF or 
any other parameter of visibility. Is it possible to 



Figure 4. The barrier flights relative to the targets {u- 
moving space). 


fly in such a way that the observer's path in u- 
moving space be of the form shown in Figure 4, but 
with 

OOi = O1O2 = O2O3 = • . . = OO'i = 0'i0'2 

= 0'20'3 = . . . = aS, 

( 7 ) 

and how are the flights to be described in the geo- 
graphical space of Figure 1? 

The answer is simple : Fly the same crossover paths 
as in Section 7.1.1, only with IF replaced throughout 
by aS. For the relation between the geographical and 
the relative paths depends in no wise on the fact that 
aS in Section 7.1.1 had the value IF. The horizontal 
bands centered on 00', OiO'i, etc., are now of width 
aS, and merely lose their meaning of “regions swept 
clean.” Moreover, all the formulas of Section 7.1.1 ap- 
ply to the present case, provided IF is replaced 
throughout by S. For convenience we shall write 
them down in the new (general) form here. 


BARRIER PATROLS 


103 


Length of upsweep for one and for n observers 
abreast spaced S miles apart, 


M = 
M = 


V -]r u 
V 


S (one observer) 
nS (n observers abreast) . 


V u 

Time of flight of one basic element, 


To = 


To = 


2L 




2L 


\/v‘^ — 


+ 


+ 


2S 

V -jr u 

2nS 

V u 


( 8 ) 


(one observer) 

{n observers abreast) . 


( 9 ) 


Time taken for first target (on left bank) not flown 
over in the flight of the first basic element but which 
will be the lowest one floAvn over in the second basic 
element to reach 0: 

2*S 

Tt = — (one observer) 

^ ( 10 ) 

Tt = (n observers abreast). 

Time of no patrolling after N basic elements (the 
last one without upsweep) are flown by n observers 
abreast, from the end of the flights until the patrol 
flights must be resumed in order to maintain the 
uniform coverage of sweep spacing S in w-moving 
space : 

No patrolling period 

= + ^. (11) 

\_u {v u) -x/ — u^j V u 


This assumes an advancing barrier, and no account 
is taken of the time for the aircraft to return to base. 

As was seen in Section 7.1.1 the barrier element 
will be of the (A) retreating, (B) stationary (sym- 
metrical), or (C) advancing types, according as 
Tt < To, Tt = To, or Tt > To respectively. Using 
equations (9) and (10) and transforming the results 
algebraically, the following criteria are derived in the 
case of n aircraft flying in line abreast at spacing S. 

Writing k = ^ 

\v — u 

the condition for 

A. Retreating barrier is n < ^ 

B. Stationary barrier is n = ^ 

C. Advancing barrier is n > k i-- 


The method of evaluating the probability P(S) of 
detecting a particular target attempting to cross the 
barrier is given in Section 2.7, equation (43) in par- 
ticular. Figure 12 of Chapter 2 shows the relation 
between this probability and S (although the abscissa 
is the sweep density “n” = 1/iS) in typical cases. If 
in particular an inverse cube law of sighting is as- 
sumed, P{S) is given by formula (47) of Chapter 2 

P(S) = erf ^0.954 

E being the effective visibility. In general, P{S) has 
to be derived by approximate formulas or graphical 
methods, in connection with the material set forth 
in Chapters 4, 5, and 6. 

It may be remarked that if the target’s speed u 
has been overestimated, the barrier flights will ap- 
pear relatively to the target not as the horizontal 
lines of Figure 3 but as two sets of parallel lines, one 
set, corresponding to flights from the left bank to 
the right, being tipped slightly down to the right, 
the other set, corresponding to flights in the reverse 
direction, tipped down to the left. Also, all upsweep 
legs will be, a trifle shortened. And thus the flights, 
while not giving as regular a picture relative to the 
target, will give one of more crowded paths, and 
hence the chance of detection will be increased. It 
will, however, not be as great as it would have been 
had the searcher planned his flights for the correct 
value of u. 

If the target is moving obliquely down the chan- 
nel, instead of exactly parallel to the banks as we 
have been assuming, the effective value of its speed 
as far as the tightness of the barrier is concerned is 
its downward component. This again tends to re- 
duce the effective speed, with the result noted above. 

There are, generally speaking, two methods of ap- 
plying the results derived here. First, the probability 
P{S), i.e., the tightness of the barrier, can be given; 
required the number of aircraft needed to maintain 
a patrol of a given sort (e.g., a symmetric one), the 
corresponding length of upsweep M (which deter- 
mines the basic element), etc. Second, given the num- 
ber of aircraft and type of patrol, find the probability 
of detection. In the first case, S is derived from P{S) 
by an equation like (13), n is determined by (12), e.g., 
(B) or (C), then M is calculated from (8). In the 
second case, S is determined from (12), and then M 
and P{S) from (8) and an equation like (13). But there 
are various mixed cases; the situation is illustrated 
in Section 7.1.3. 


> 


(13) 


104 


THE SEARCH FOR TARGETS IN TRANSIT 


We are now in a position to solve the problem, 
foreshadowed in Section 7.1.1, of the use of an ad- 
vancing element barrier to avoid 24-hour flights. Sup- 
pose that for A hours out of the 24 it is expedient to 
fly the patrol, whereas during B hours it is inex- 
pedient; B might be the hours of darkness in cases 
where much dependence is placed on visual detec- 
tion or recognition; A + 5 = 24. Let us try to de- 
termine the number n of aircraft patrolling abreast 
and the number N of circuits they must patrol to- 
gether so that coverage at the sweep spacing S in 
w-moving space be maintained constantly. For any 
given N and n, the time of no patrolling is given by 
(11), while the total patrolling time is A^To — M/v 
(the —Mjv term, because the last upsweep is 
omitted). The sum of these two periods of time 
must equal 24 hours; using (8), (9), and (11), one 
finds that 

= ^. (14) 


Now since the permissible no-patrolling time must 
be at least as great as B, (11) leads to the inequality 


N 




2vnS 


2L 


{V + U) 


nS 

V u 


^ B. (15) 


P{S) [by the use of (13)]. This last involves some 
trial and error; moreover, a value of S which has 
these properties may turn out to impose too great 
force requirements, and so one may have to be satis- 
fied with a somewhat lower probability P{S), i.e., use 
a larger S. When the value of S has finally been 
chosen, (14) gives w as a positive integer, and the 
problem is solved. It will be illustrated in Section 
7.1.3. 

7.1.3 Practical Applications 

The following examples are illustrative: 

1. It is desired to close a 600-mile channel by a 
barrier giving a 90 per cent chance of detection. The 
speeds are v = 130 knots and u = 12 knots. The 
effective visibility is 20 miles, and equation (13) is 
assumed.^ How many aircraft are needed in order to 
have a symmetric element barrier, and how should 
the flights be specified? 

Using (13), erf (0.954 X 20/*S) = 0.9, and it is 
found from a table of error functions that erf (1.163) 
= 0.9, hence 0.954 X 20/S = 1.163, so that S = 16.4. 
Since the inequality (A) of (12) must not hold, the 
number n of aircraft must be the smallest ' integer, 
not less than 


By multiplying this through by A(i; + tt), introduc- 
ing the quantity k of (12), and replacing nN by its 
value given in (14), one derives the quadratic in- 
equality 

2LkN^ — {Av — Bu)N — 12u ^ 0, 

which (on completing the square, etc.) is readily 
shown to be equivalent to the following: 


N ^ — .Bu + a/ {Av — BuY + 96LA:wJ. 


(16) 


L u jv u _ 600 X 12 /l42 _ o 7 

S’V \lv - u ~ 16.4 X 130 VlTS “ 

in other words, four aircraft are necessary. But with 
four aircraft, case (C) of (12) is in effect, not case (B) : 
The barrier advances up the channel. To have a 
stationary one, the four aircraft may fly closer to- 
gether than S, by an amount determined by solving 
the equation 

600 X 12 /T42 
130*S VllS’ 


It is noticed that this inequality does not involve S. 
And now we have the conditions (14) and (16) in all 
respects equivalent to (14) and (15) (one set is a 
necessary and sufficient condition for the other): 
Thus (14) and (16) are the necessary and sufficient 
conditions which N and n must satisfy to be solu- 
tions of the problem. They do not, however, fully 
determine N and n. The method for doing this is 
as follows: First, choose for N the largest integer 
satisfying (16). Second, choose a value for S which 
on the one hand makes 12u/NS an integer, and on 
the other hand gives an acceptably high value to 


i.e., equation (12), case (B). We obtain S = 15.15 
miles. With this reduced spacing, the barrier gains 
in tightness; in fact the probability found from (13) 
now becomes a 92.5 per cent chance — all to the good. 
The length of upsweep given by (8) is M = 55.5 
miles. Finally, the lead angle given by (1) is a = 5° 18'. 
These quantities determine the fundamental ele- 
ment, or rather elements, as four congruent sym- 
metrical crossover paths flown shown in Figure 5. 
But in determining force requirements it is not 

®^The value of 20 miles is too low for ships but is about cor- 
rect for surfaced submarines. 


BARRIER PATROLS 


105 


sufficient to have only the fundamental flights given; 
we must find how long their flying takes and con- 
sider questions of aircraft endurance. It is found 
from (9) (with n = 4, >S = 15.15, etc.) that Tq = 
10.14 hours. Now only a long-range patrol aircraft 
such as a PBM or PB4Y could have an endurance 


144/aS = 9 and P{S) given by (13) is a probability 
of nearly 91 per cent. Then n = 9: nine aircraft must 
be flown spaced at 16 miles abreast, one flight only 
being made per day. The time of patrolling is Tq — 
M/v (the last upsweep being omitted) which has the 
value [given by (8) and (9)] of 1034 hours, the re- 



OP : PQ: QRsIS.IS Ml 
0'0"»P’P"»Q'Q" = R'R"=55.5 Ml 
00':600MI 

Figure 5. Symmetrical barrier flown by four aircraft abreast. 


sufficient for this one circuit of the fundamental 
element; it will have to be capable of well over 10 
hours, since time must be allowed for investigation 
of contacts, the trip to and from base (which may 
not be at 0, P, Q, R) etc. And it cannot be expected 
to make more than one circuit. Thus a new flight 
of four fresh aircraft must be readied and waiting 
at 0, P, Q, R in order to take up the flights as soon 
as the first set return to these points. The operation 
will therefore require eight aircraft, assuming main- 
tenance to be quick and perfect. Actually, a few 
more should be on hand, as well as enough pilots and 
lookouts to ensure their being well rested at the out- 
set of every new flight — an essential condition for 
their efficient operation, without which the effective 
visibility will fall far short of the assumed 20 mile 
figure. 

2. Under the assumptions of the last example, let 
it be required to fly a barrier of the advancing type 
with n aircraft abreast during the A = 12 hours of 
light, to be discontinued during the P = 12 hours of 
darkness. Applying the method at the end of Section 
7.1.2, we obtain from (16) that N ^ 1.16, and hence 
we take N = 1. Next we must take an S which gives 
l2u/NS = 144//S an integral value and provides an 
acceptable projiability P(>8), while at the same time 
not using an undue number of aircraft. We have seen 
that the value 16.4 gives a 90 per cent chance of 
detection, and this suggests taking P = 16, for which 


maining 13^ hours (including the 12 of darkness) not 
needing any patrol. The fact that the two periods are 
not each equal to 12 hours is of course due to the 
circumstance that since N and n are integers, (14) 
and (15) cannot be solved as equations but rather 
(14) as an equation and (15) as an inequality. The 
operation thus requires about the same forces as in 
the previous example. Which of the two methods is 
to be used depends on considerations of equipment 
(how good the radar is for search at dark, etc.) and 
tactics. 

3. A channel 300 miles wide is to be barred by a 
symmetrical barrier flown by one aircraft of 6-hour 
endurance. How tight is the barrier, and how fre- 
quently does the aircraft have to be relieved? Assume 
again u = 12, v = 130, E = 20, and equation (13). 

From equation (12), case (B),with n = 1, we obtain 
S = kLujv — 30.4. From equation (13), P = 0.644: 
a 64.4 per cent chance of detection. Eliminating P 
from equations (9) and (12), case (B), we have 


V 

which, in the present case, gives Tq — 5.06 hours. 
Evidently with an endurance of 6 hours, just one 
circuit can be flown, so we shall require about five 
flights a day, and between two and five aircraft 
available at the very minimum. 


106 


THE SEARCH FOR TARGETS IN TRANSIT 


^ ^ ^ Barrier When Target Speed Is Close 
to Observer’s Speed 

So far it has been assumed that v considerably ex- 
ceeds u] indeed, when v ^ u, the crossover type of 
barrier is kinematically impossible. This is no ob- 
stacle when the observer is airborne and the target 
is a ship, but when both observer and target are 
units of the same type (both ships or both aircraft), 
the situation excluded heretofore becomes important. 
Although many plans of barring a channel can be 
devised for this case, attention will be confined here 
to the very simple case in which the observer moves 
back and forth across the channel on a straight path 
perpendicular to its (parallel) banks: such a patrol 
is always possible and its design evidently does not 
involve the speed ratio u/v. 

This back-and-forth barrier will be compared 
with the symmetrical crossover (when u < v), and 
since only a rough comparison is sought here, the 
definite range law will be assumed (range = in 
each case). A more accurate detection law is not 
likely to alter the comparison appreciably. 

The two diagrams on Figure 6 show the geographic 
as well as the relative tracks for the two types of 
patrol. 

In each relative track a half cycle has been se- 
lected and the area swept shaded. The probability 
of detection for each case has been taken as the ratio 
of the shaded area to the total area in the channel 
between the two dashed lines marking off the half 
cycle. It is convenient to introduce two new vari- 
ables to describe the probability of detection, r = vju 
and X = L/ IF. For the case of the crossover patrol, 
the probability Px of detection is given by 



For the back-and-forth patrol the probability P 
of detection is given by 

^ fi - (x - 1 Y/x(x+i) 

~ I ^ for r ^ 2VX(X + 1) 

U for r > 2\/X(X + l). 

In Figure 7 the values of P for the two cases are 
plotted as functions of r with X kept fixed for a given 
curve. In comparing crossover patrols with back-and- 
forth patrols, curves bearing the same value of X 



sin<r»7 Ltoncf 

S =(1 + 7) Lton CT 



Figure 6. A comparison of barriers. 




CIRCULAR BARRIERS 


107 


should be compared. The solid curve passes through An example will illustrate the use of the curves, 
the points of intersection of the curves being com- Suppose a ship making 12 knots is trying to pre- 
pared and marks the boundary between the regions vent undetected penetration of a barrier by a 6-knot 

1 

0.8 

0.6 

P 

0.4 


0.2 

0.1 

012 34 56789 10 

Figure 7. The comparative effectiveness of back-and-forth and crossover plans. 


r* i.l 

Xro.24 x.o 


/ 

/ 

'K '' 

A7 

\ 

\ 

z 

/ 

/ 






s\V -4 
V 

^ /‘ 
/ 

/ 

> 

'-'A 7f '' / 

/ \/i / ' 

1 y 


/A- 






_ y'' 

i7\i 

’'/ A 
'// A 

Vy ✓ ^ 

/-►A' 

V 







' vA''" 

y/ 

\ 

^ A > 




— 

Pm 

— 














where back-and-forth is preferable and where cross- 
over is preferable. 

In order to facilitate the selection of the preferable 
type of patrol, Figure 8 is included. This curve shows 
the relation between X and r for the points of inter- 
section of curves in Figure 7. 



0 12 3 4 

Figure 8. Regions of effectiveness of back-and-forth 
and crossover plans. 


submerged submarine. Assume further that the chan- 
nel being guarded is 8 miles wide and that the sonar 
search width IF is 2 miles. Then L = 7 — 2 = 5, 
X = 5/2 = 2.5, r = 12/6 = 2. Entering Figure 8 with 
these values for r and X one discovers that a cross- 
over patrol is preferred. 

72 CIRCULAR BARRIERS 

^ ^ ^ Constant Radial Flux of Hostile Graft 

In the case where enemy surface craft or sub- 
marines are attempting to leave a point of the ocean, 
such as an island or exposed harbor, and in the case in 
which they are attempting to approach such points 
or to close positions at which our forces are conduct- 
ing landing operations, the vector velocity pattern 
is a radial one: it is “centrifugal” (directed away from 
the central point) in the former case and “centripetal” 
(directed toward the center) in the latter. But in 
each case it can be regarded as constant in time : over 
long periods, the density of outgoing or incoming 
craft is not expected to vary. This sets the present 
situation in strong contrast with that considered in 
Section 7.3, in which the unit to be detected is, to be 




108 


THE SEARCH FOR TARGETS IN TRANSIT 


sure, proceeding radially away from a point of fix, 
but in Avhich its likelihood of being at various dis- 
tances from this point depends strongly upon the 
time elapsed since the fix. Corresponding to this 
latter circumstance, the layout of the search provides 
for a progressive variation of the searched positions 
with lapse of time; the theory of such plans is far 
more complicated than that of the ones considered 
at present. 


^ ^ ^ Targets Moving Toward a Central 
Objective 

The type of operation considered here may be con- 
trasted with that of protecting a convoy against sub- 
marine attack considered in Chapters 8 and 9. The 
defended position is in the present case at rest, and 
thus the submarines have no tracking problem and, 
moreover, can approach at any speed at which they 
find it convenient to operate, and from any relative 
bearing. The defended position is consequently ex- 
posed to attack in any direction from which it is not 
effectively shielded by land masses or shoal water. As 
in the case of convoy escort, the protection given will 
be of two kinds. Aircraft will engage in barrier pa- 
trols outside distances within which wholly sub- 
merged approach is possible (more than 60 miles out 
with the submarines of World War II). An inter- 
mediate aircraft screen may be provided to pick up 
submarines which may be surfaced at distances un- 
der 60 miles, but this screen will be less important. 
An inner screen of surface ships will patrol a barrier 
for submarines (submerged or surfaced) which may 
elude the aircraft screen. Details of each screen 
follow. 

It is advantageous to abandon the design de- 
scribed in Section 7.1 in favor of a simpler plan. The 
lead-angle a which was a key element of the basic 
design given in Section 7.1 is required only because 
of the necessity to search parallel strips of space 
moving with enemy velocity u, patrolling with own 
velocity v, when the direction of search was neces- 
sarily opposed in alternating members of each pair of 
strips. If search of adjacent strips is always carried 
out in the same direction there is no need to use a 
lead-angle at all. The resulting searched strips in u- 
moving space will still be parallel but they will be 
inclined to the target track at an angle (90° — 
tan“i u/v) instead of at 90 degrees. 

When the barrier is set up around an objective the 


closed barrier path may be made to contain the 
objective, and the lead-angle a can become zero. The 
resulting path is a simple circle, or more practically, 
a square. Such a path is much easier to navigate, or 
to evaluate, than a set of barriers set up on barrier 
lines which form the sides of a polygon. 

The size of the square will usually be determined 
by tactical considerations, i.e., the distance at which 
interception should be achieved. For example, against 
submarines this may signify the distance within 
which approach while wholly submerged becomes 
possible. The distance the target can travel while 
the searcher makes one circuit, or between sweeps 
by equally spaced searchers on the same square 
track, becomes the track spacing S in i^-moving 
space. For the simple case of a circular track this is 

S = " (18) 

71 V 

where r is the radius, and n the number of search 
craft employed. Since with a given effective visibility 
there is a contact probability corresponding to any 
value of 8, formula (18) determines the number of 
searchers required to give any desired tightness. 

For the more practical case of a square track of 
side h 


46 u 
n V 


(19) 


This value of S corresponds to the minimum contact 
probability since with a square the target track will 
not always cross the search track at 90 degrees. 

There is one important difference between this 
type of barrier and certain cases of the crossover 
type. 8 for a given size of square is determined wholly 
by relative target movement and the number of 
equally spaced searchers (8 is proportional to ujuv), 
and no explicit choice as to its value determines the 
search path. In this respect it is like the continuous 
symmetrical crossover patrol. This raises the ques- 
tion as to how two searchers patrolling abreast should 
be spaced. If the definite range law is applied, search 
abreast at a distance apart equal to twice the range 
would be equal in effectiveness to search by each 
singly, equally spaced on one and the same square 
path. With any other kind of contact probability 
law the corresponding track spacing in search abreast 
is the value of 8 which is found by using formula (19) 
and the appropriate value of n. It will be noted that 


CIRCULAR BARRIERS 


109 


this is not equivalent to placing two or more search- 
ers closely abreast on exactly the same track. They 
must be either equally spaced on an identical track, 
or, if abreast, spaced at right angles to that track 
by exactly the right amount. The effects of irregulari- 
ties in spacing are, however, rather small. It should 
be noted that if the value of S/n is at all large, search 
abreast on squarq or circular paths cannot be car- 
ried out effectively because of the difference in the 
length of track for each observer. 

It is apparent that the maximum size of square is 
fixed by (a) the endurance of the searcher (endur- 
ance > 46) and by (b) the number of searchers con- 
tinuously available. When surface craft are em- 
ployed against underwater targets it is very desir- 
able that two or more patrol abreast. Since a large 
number will be required to cover a square of any 
size, it will be more practical to employ small groups 
(three to six) in crossover patrols on barrier lines 
which form a polygon with a convenient number of 
sides, rather than have them all patrol a single large 
square. 

7.2.3 Patrols Against Incoming 

Submarines 

Detailed application of the foregoing considera- 
tions to a particular situation depends mainly on the 
range of detection to be expected from the radar gear 
installed on the aircraft. This will determine the in- 
terval at which aircraft patrol on the basic square 
track. The longer the radar range the fewer aircraft 
need be employed, and the easier it will be to navi- 
gate along the track. With S-band or X-band radar, 
navigation will be made very easy by the constant 
presence on the screen of check points on the island. 

The distance from the position being protected 
from submarines at which the patrol track will be 
placed will be the estimated submerged run of sub- 
marines (not over 70 miles) plus the effective vis- 
ibility for the radar used, as given in Chapter 5. With 
ASG radar used on patrol planes in World War II 
this will result in flying a square with legs of about 
180 miles to each side. With PBM aircraft the com- 
plete circuit of the square will then require 6 hours. 
Two complete circuits can be made by each sortie 
in 12 hours. Reference to formula (19) and standard 
tables of effective visibilities shows that only two 
120-knot aircraft need patrol at one time equally 
spaced in order to give a 50 per cent chance of con- 


tact with a 15-knot surfaced submarine. Four air- 
craft will give a probability of contact of 77 per cent 
and six aircraft will give a probability of contact of 
about 90 per cent. With the older and less effective 
ASB radar (E = S miles), however, six 150 knot air- 
craft patrolling a square with 150 mile sides will only 
give a probability of contact of 70 per cent, and each 
sortie will be limited to a single circuit (four hours). 
Thus 18 TBF sorties give less protection than four 
PBM sorties. 

Consideration of these figures leads to the con- 
clusion that PBM aircraft should be used where 
available. A single squadron can probably offer six 
aircraft a day, at least for a limited period. These 
can be divided between four by night and two by 
day, to give a nearly uniform probability of contact 
of about 77 per cent, if due account is taken of the 
added possibility of sighting submarines visually in 
the daytime and the need to maximize the defensive 
value of the patrol. The probabilities given are ap- 
plicable to certain types of Japanese submarines of 
World War II. 

The force requirements can be reduced by pa- 
trolling smaller squares, but it is questionable whether 
this expedient should be employed. There is no gain 
in security if we give up the outer screen in order to 
patrol waters through which submarines are likely to 
transit largely or entirely while submerged. Screen- 
ing operations within fifty miles of the beach are 
largely the responsibility of surface craft, except at 
night. Where additional forces are available, such as 
short range carrier-based aircraft, they may well be 
employed in patrolling an additional smaller square. 
This is particularly important at night when the sur- 
face screen may become less tight, owing to the 
possibility that the submarine may surface, and thus 
make higher speeds. Patrol of a square only 50 miles 
from the beach by only three TBF aircraft, even at 
night, gives an 80 per cent probability of contacting 
any surfaced submarines which may slip by an outer 
screen of four PBM’s. This raises the overall prob- 
ability of contact at night to over 95 per cent. 

When the land mass involved is unsymmetrical 
and only one dimension is larger than 20 miles (a 
long, thin island), the square patrol may still be the 
simplest and most efficient method of setting up the 
antisubmarine barrier, when the landing operation 
being defended is made at or near one end of the 
island, even though a small part of the flying is over 
land. When all dimensions are large a barrier patrol 
or a combination of barrier patrols of the familiar 


no 


THE SEARCH FOR TARGETS IN TRANSIT 


crossover type will be set up at the appropriate dis- 
tance from our forces, along the coast. 

^ Surface Patrol Against Incoming 
Submarines 

Surface craft on inner screens are assigned the 
responsibility of catching submerged submarines 
which may have eluded the outer aircraft screen. 
This screen will be placed outside torpedo range, but 
not so far outside as to require excessive patrol craft 
or as to thin out the tightness of the screen. Unless 
the patrol line has other than antisubmarine func- 
tions, it is therefore unlikely that it will be set up to 
surround an entire island, and the barrier patrol will 
be composed of straight line segments each of which 
follow an appropriate design based on an obvious 
application of the principles made familiar in earlier 
parts of this book. 

In all cases, even when the size of the island per- 
mits patrolling all the way around it, it is very im- 
portant to distribute the forces available so as to 
sweep as wide a swath as possible, rather than to have 
single ships or pairs of ships patrolling in column, or 
in short segments of the patrol line. The submarine 
can easily evade one or two ships; in addition, when 
ships patrol a very short segment there is the serious 
difficulty of continual interference by own wakes. 
At least three ships, and preferably four to six ships, 
should always patrol abreast. Under normal sound 
conditions, and using plans of this type, a group of 
four ships patrolling abreast can hold a very tight 
line almost 40 miles long. With the forces normally 
assigned to such operations, a line 200 miles long is 
easily maintained. The required patrol line is usually 
less than 100 miles long, and ten to twelve ships dis- 
tributed between two or three groups should be more 
than sufficient. Additional forces usually available 
may be used to strengthen this line still further or to 
set up additional screens at other distances. 

7.2.5 pg^trol Against Centrifugal Targets 

The general considerations concerning the circular 
(or square) barriers, as well as the closer barriers 
under certain circumstances, set forth in Sections 
7.2.2, 7.2.3, and 7.2.4, above apply in an obvious 
manner to the case of targets seeking to leave the 
central point. No further discussion is required here. 


7 3 SEARCH ABOUT A POINT OF FIX 

^ ^ ^ The General Question 

When an object of search on the ocean, such as a 
surface craft or downed airplane life raft, has had its 
approximate position disclosed to a searcher at a 
certain time, the searcher has the problem of dis- 
posing its subsequent searching effort (which is al- 
ways limited) in such a manner as to maximize its 
chance of detecting the object, subject, of course, to 
the practical restrictions of navigation. The informa- 
tion regarding the object’s approximate position may 
be derived from a DF fix, the report of a chance ob- 
servation, by indirect inference, or, in the case of the 
life raft, from a radio communication from the air- 
craft about to crash. The point at which this informa- 
tion locates the target is called the point of fix and 
the time for which the information is given, the time 
of fix. It is assumed that the searcher is airborne, and 
thus has a considerable speed advantage over the 
target. 

If the fix were a perfectly accurate one and the 
target were at rest or moving in a known manner, the 
searcher’s task would be simple. He would proceed to 
the point of fix in the former case, and would search 
the locus of points to which the target, initially at 
the point of fix, could be assumed to have moved 
during the intervening time in the latter case. But 
such accuracy of fix is seldom if ever obtained : Only 
a probability distribution of target positions at the 
time of fix is actually given. This distribution mil 
have its greatest density at the point of fix and fall 
continuously to zero at a distance. In an important 
group of cases, this distribution can be regarded as 
symmetrical about the point of fix and can indeed 
be taken with satisfactory accuracy as a circular 
normal one, 

f(x,y) = fir) = (20) 

where f{x,y)dxdy is the probability that the target 
at the time of fix be in the small region dxdy at the 
point {x,y) at the distance r from the origin 0 (which 
is at the point of fix) and where a is the standard 
deviation. 

It should be remarked that DF fixes usually do not 
give rise to circular distributions, but under certain 
conditions the distributions to which they lead are of 
this character. 

Two cases are considered in this chapter. In the 


SEARCH ABOUT A POINT OF FIX 


111 


first, the target’s motion is negligible, so that it can 
be regarded as at rest; equation (20) gives its dis- 
tribution at all subsequent times. In the second, the 
target’s speed is known but its direction is not but 
is assumed to be uniformly distributed in angle 
throughout the full circle; the distribution after the 
lapse of time t after the fix has already been derived 
in Chapter 1, equation (10). The solution in the second 
case will be derived more or less directly from the 
first. It is to the second case that the mathematical 
schema of the centrifugal vector field mentioned at 
the beginning of this chapter applies exactly, but in 
contrast to the cases of Section 7.2, the density of 
targets is not constant, but after being humped up 
about the center spreads itself out into a thick ring 
cut normally by the vectors, with the lapse of time. 

Square Search for a Stationary Target 

In this case, as we have seen, equation (20) gives 
the probability density of the distribution of the 
target for all later time. If the total searcher’s track 
length during the search is L miles and his search- 
width IT, then the quantity of searching effort as it 
has been defined in Chapter 3 is <I> = WL. The 
problem of so disposing a continuous spread of 
searching effort o^ total amount ^ that the prob- 
abihty of detection is maximum has been solved in 
Section 3.4. But here we are confronted by the prac- 
tical problem of designing actual navigable flights 
which will maximize the probability of detection. 

The type of flight which it is expedient to use 
consists of a set of “expanding square” flights of the 
sort shown in Figure 9. After passing over the point 
of fix 0 to a point S miles beyond 0, the aircraft 
turns through a right angle (e.g., to the right), and 
after S miles it turns again through a right angle in 
the same direction, continuing 2S more miles before 
turning a third time; after flying 2S miles it turns 
again, and continues in this manner, always keeping 
adjacent parallel tracks S miles apart. After a certain 
number of legs have been flown, there results an ap- 
proximately square figure covered by equally spaced 
lines, the space between them being the sweep spac- 
ing S. Such a figure will be called a “square of uniform 
coverage.” The underlying scheme of the search is to 
fly a succession of superimposed squares of uniform 
coverage, each centered at 0, and of successively 
large dimensions. This will furnish a practicable 
means of approximating to the theoretically optimum 


continuous distribution of searching effort derived in 
Chapter 3. 

The first problem is to determine a desirable value 
of the sweep spacing S, up to now left arbitrary. The 
point of view adopted here is that S should be so 
chosen that on the initial square the probability of 
detection per unit time shall be a maximum (during 
the important part of the search, i.e., the beginning). 
Clearly such an S will, for a given law of detection, 
be a function of the parameters of detection and of 
the standard deviation of the distribution a. In the 



case of the inverse cube law of detection, S will be a 
function of E and a. Since it can be shown that S 
will not be sensitive to the law of detection, it is 
permissible to assume a convenient one. We shall as- 
sume the inverse cube law, in which, as was seen in 
Chapter 2 [equations (26) and (46)], 

p{x) = 1 - exp j^- 0.092 , 

where p{x) is the probability of detection of a target 
of lateral range x from a straight aircraft track. 

Consider the distribution (20) of targets before any 
flights are made. The probability of the target lying 
on the strip parallel to the y axis between x and 
X + drr is found (by summing probabilities) to be 


112 


THE SEARCH FOR TARGETS IN TRANSIT 


dx f " f{x,y)dy = — ^ (21) 

J -CO (TV 27r 


The graph of the differential coefficient against x is 
the familiar normal law curve, reaching its maximum 
at the origin. Now suppose that an indefinite straight 
flight is made along the y axis and has failed to 
detect the target; in the light of this additional 
knowledge, the distribution of targets is altered, and 
the differential coefficient in (21) no longer repre- 
sents the lateral density of targets [i.e., the prob- 
ability in the {x, x + dx) strip]. To find the new 
density, the use of Bayes’ theorem is called for (see 
Section 1.5, footnote c). The “a, priori probability” 
of the target’s being in the (x, x + dx) strip is given 
by (21); the “productive probability” of the event 
(viz., of not detecting the target on the sweep through 
0) is 

1 — p{x) = exp ^ — 0.092 

and hence the “a posteriori probability” density of 
probability of the target’s being in the (x, x + dx) 
strip is proportional to the product 

/ a:2 0.092E‘^\ 

which has its maxima at x = +0.65 \/E(t', it is no 
longer humped at the origin but presents double 
humping with an intervening depression at the origin. 

Where must a second indefinite sweep parallel to 
the y axis be made if it is to achieve the greatest 
probability of detecting the target? Since the dis- 
tribution obtained above is skewed, the distance D 
between the first and the second track should be 
slightly greater than the distance 0.65 \/ Ea out to 
the maximum of the new distribution. To find it 
precisely, we compute the probability of detection by 
multiplying the expression (22) with p{x — D) and 
integrating over all positive x. It appears at once that 
in order to maximize this probability, D must make 
the function 



fo 


0.092£;2 0.092^;2l 

:r2 (x - Dy\ 


dx 


a minimum. By trial, it is found, in using numeri- 
cal integration, that the approximate value of D is 
0.75\/E<t. 

Clearly, if the searching were done by means of 


infinite parallel equispaced sweeps, the sweep-spacing 
S would be taken equal to D if the chance of early 
contact is to be maximized. The square search of 
Figure 9, while not exactly of this type, is near 
enough so that an obvious choice of S is to give it 
the same value D; accordingly, we shall use hence- 
forth the sweep spacing 

S = 0.75\/^. (23) 

Returning to the square of uniform coverage, it is 
seen that the three-circuit flight path of Figure 9 can 
be inscribed in a square of side 2 X 3*S. More gen- 
erally, an N-circuit flight of this sort can be inscribed 
in a square of side 2NS. The total length of tract 
from 0 to F is found (as an arithmetic progression) 
to be L = 2iS X 2N {2N + l)/2. Hence the average 
density of searching effort (the </> of Section 3.3) is 
TFL/area, or 


WS 


2N(2N + 1) 
{2NS)^ 



W 

'S' 


which is approximately W /S, the value to be adopted 
here. 

Consider now a sequence of n squares of uniform 
coverage centered at 0 and of half-side Sk {k = 1, 2, 
• • -,71), where 

0 < Si < S2 < S3 < * • • < Sn. 

If {x^y) is a point in some of these squares, let us 
say in those of side s„i _|_ 1, s„j 4. 2, • • • , s„, then the 
mean density of searching effort performed by these 
squares is {n — m) W/S. Thus the flights give rise 
to a searching effort function z = <(>* {x,y), the graph 
of which (in xyz space) is of the form of a tapered 
heap of square slabs of thickness W/S, piled 
upon one another and centered on the z axis. The 
total volume of this pile, j j 4)^ {x,y)dxdy, must equal 
the total searching effort $ = IFL; thus we must have 

n 

4 XI s*' = SL. (24) 

k = l 

It is by means of this function 4>^{x,y) that we must 
approximate the solution 4>(x,y) obtained in Section 
3.4 of Chapter 3. There it was shown that, outside the 
circle of radius a given by 

a* = 4<r2 i , (25) 

IT 

4>{x,y) is zero, while within this circle, it is given by 


SEARCH ABOUT A POINT OF FIX 


113 


<f>ix,y) = 0(r) = (26) 

The graph of z = <t>{x,y) is thus a paraboloid having 
the z axis as axis of revolution, cutting the xy plane 
in the above circle, beyond which the paraboloid is 
replaced by the xy plane. 

Thus, graphically put, our problem is to determine 
the quantities n, Si, • • •, subject to (24), so that 
the piled slab solid shall approximate the parab- 
oloidal one. The heights of the two solids being 
nW/S and a^l2a‘^ respectively, n is determined by 
equating them, 


Since n must be an integer, (27) means that it must 
be taken as the nearest integer to the right-hand 
member. To find s*, consider the space between the 
two horizontal planes z = — {k — 1)TF/>S and 

z = aV2o-2 — kW/S; they contain the kth slab of 
the piled slab solid, and hence cut from it the volume 
/S] and they cut from the paraboloid a volume 
of revolution readily found by integration to be 
Tr{2k — 1) cr’^W'^/S'^. On equating the two volumes, 
we obtain 

= T ¥ - 1), fc = 1,2, (28) 

Thus the volumes of the piled slab solid and the 
paraboloidal solid are equal [and hence (24) is auto- 
matically satisfied, since it expresses the required 
equality of this volume with 4> = TTL, a requirement 
met by the paraboloidal volume, according to Sec- 
tion 3.4.], and the two solids agree in position about 
as closely as possible. 

The number and dimensions of the squares of uni- 
form coverage are determined by equations (27) and 
(28) ; but except for the fact that they are all centered 
at 0, their positions (relative orientation) have been 
left arbitrary. We now lay down the following rule: 
The second square should he tipped so that its side 
makes J^5 degrees with the side of the first, the third 
should similarly he at 45 degrees with the second {and 
thus he parallel to the first) and, in general, the {k 1) 
should he at 45 degrees with the kth [and parallel to 
the {k — 1)]. 

The justification of this rule is twofold: 

First, a greater randomization of flights is achieved ; 
i.e., there is less danger of passing over the same path 
twice in succession, with resulting loss of efficiency. 


The situation in this regard is illustrated by the 
following considerations. Navigational errors will not 
be apt to permit the second search to be flown along 
the optimum tract which is approximately midway 
between and parallel to the legs of the first search. 
If p{S) is the probability of detection with sweep 
spacing S, then the probability of detection with two 
searches when the optimum track on the second 
search is attained is p{S/2). However, if the second 
search duplicates the first search, the probability of 
detection with two searches is p(S/\/2), (assuming 
the inverse cube law). If the legs of the second search 
are inclined to those of the first search, so that the 
probabilities of detection on the two searches may 
be considered as independent, the probability of de- 
tection with two searches is 1 — [1 — p(S)Y. In gen- 
eral this latter probability will be slightly less than 
piS/2) but considerably greater than p(>S/\/2). For 
example, using the inverse cube law with E/S = 0.1, 
we have 

p{S/2) = 0.212, 
p{S/V2) = 0.151, 

1 - [1 - j5(S)]2 = 0.203. 

Thus, there seems to be more to gain than to lose by 
inclining the legs of the second search to those of the 
first search. 

Second and more important, the {k 1) square 
flown according to this rule will sweep a maximum of 
important unswept water: consider the situation 
after one square of uniform coverage has been flown 
without resulting in detection; the area which it is 
most important to sweep is the part of the circle 
circumscribing the late square but outside the latter. 
With the next square tilted at 45 degrees, a maximum 
of this area is covered. Moreover, with the tilted 
square the region of overlapping of two successive 
squares is least. 

It remains to evaluate the probability P* of de- 
tection by the square search described herein, and to 
compare it with the probability P for the idealized 
search of Section 3.4. It is shown that a lower value 
of P* is given by 

p** = (1 -e-w/s'^g-wn/s ^ gWk/s (2k - 1), 

k=l 

(29) 

while P is given by 



114 


THE SEARCH FOR TARGETS IN TRANSIT 


These two functions are plotted in Figure 10 for 
various values of n; the P** curves are in solid line 
and the P curves are dotted. The P* curves lie be- 
tween these curves. 

Evidently, P* will be less than the probability P 
for the idealized search, since P is the maximum 


tribution before the squares are tilted, i.e., when they 
are all parallel, and let P** be the probability of de- 
tection with this distribution of effort under the as- 
sumptions of Chapter 3. Then P* will be greater 
than P**, for reasons already given above. By 
formula (8) of Chapter 3, the probability that the 



s 

Figure 10. Probability of detection. 


probability which can be obtained with the given 
amount of searching (provided the assumptions of 
Chapter 3 are made). 

Using the target distribution as given in formula 
(20) and the density of searching effort as given in 
formula (26), the probability P of detection for the 
idealized case is obtained from formula (15) of Chap- 
ter 3 as follows: 

^ f{x,y)dxdy, 

A 

= ^2j^2Tre-'’/2»* (1 _ g-(a*-r=)/2,=) 

Using formula (27), P may be written in the form of 
equation (30). 

The direct computation of P* is difficult because 
each square is tilted 45 degrees to the preceding 
square and each square, except the first, is not in- 
cluded completely in the following square. To ob- 
tain a lower limit to P*, consider the square dis- 


target will be in the square bounded by a: = ±Si, 
y = ±si, and will be detected by the n coverages 
of the square, is 

I T ' (1 - e-<-^'+y'^/^'dxdy 

Zira^jQ Jo 


= (1 


e-nr/S) grP 

(TV TT 


Similarly, the probability that the target will be in- 
side the square bounded by a: = + y = ± Sa: but 
outside the square bounded bya: = ± 1 ,?/ = ± Sa:- i 

and will be detected by the n — k \ coverages 
of this area is 


P, = (1 - 1)^/.) (erf2 - erP 

Thus, P** is given by 

P**=j^Pk = 


k=l 


Using formula (28), P** may be written in the form 
of equation (29). 






SEARCH ABOUT A POINT OF FIX 


115 


Retiring Square Search for a Moving 
Target 

The case here* considered is that in which the 
initial distribution at the time of fix is given by 
equation (20), but with the target moving with an 
estimated constant speed it in a random direction. 
This is the second case mentioned in Section 7.2 and 
corresponds with what can be expected to occur when 
the fix has been made on a moving target (direction 
unknown) by a method of observation which has not 
imparted information to the target and thus in- 
fluenced its motion. The problem of finding the dis- 
tribution at the later time was solved in Section 1.6, 
where it was found that the probability density /(r,0 
at a point r miles from the point of fix and t hours 
after the time of fix is given by the equation 

T ■ 

The function f{r,t) has been plotted for various 
values of ^ as a function of r in Figure 17 of Chapter 1. 
(These curves are cross sections of the distribution 
surfaces in a vertical plane through the point of fix.) 
It will be seen that when ut is at least as great as 3(7, 
the distribution has its maximum at approximately 
r = ut. Moreover, the shape of the curve for the 
large values of t is approximately the same as the 
shape of the curve of the initial distribution. These 
statements can be verified by using the asymptotic 
approximation 





for large values of x. The asymptotic approximation 
for f{r,t) then becomes 

g-(r - «0V2<r* 

277(7 a / 2Trutr ’ 

for large values of utrla^. When r is close to ut, the 
approximation becomes 


struct a search for large values of the time T which 
has elapsed from the time of fix to the initiation of 
the search. In order to obtain the maximum prob- 
ability of detection per unit time, the first circuit 
should be flown on the peak of the distribution (the 
peak is circular — it is the top of a ring); the second 
circuit a distance S as given in equation (23) inside 
or outside — say outside — the peak; the third, a 
distance S inside the peak; the fourth, a distance 2>S 
outside the peak; etc. Since the peak of the distribu- 
tion moves out approximately at the speed of the 
target, the ideal track on each circuit is an equian- 
gular or logarithmic spiral. 

We shall approximate each circuit by four legs 
with 90-degree turns, as shown in Figure 11. Let 


Le 


B Li 


Li 

A 


•'2 

A '^3 

L3 

* v: 

y 

u 

◄ 

u 



Figure 11. Retiring search. 

the lengths of the legs be Li, Lz, etc., and let the 
corresponding distances of the legs from the point of 
fix be ri, r 2 , ra, etc. The time required for the air- 
craft to fly from A to B is (r 2 + n)/?;, whereas the 
time required for the target to move from distance 
ri to distance r 2 from the point of fix is (r 2 — ri)/u. 
The aircraft will just keep up with the target if these 
times are equal. In this way we find that 


e- (r - Ut)y2a^ . 

fir.t) ^ 7 =: , — large, r — ut close to zero. 

2irffUtV2ir 

The statements concerning the curves become ap- 
parent from this latter approximation. 

From the statements made in the last paragraph 
and the results of the preceding section we can con- 


where 


r2 = mri, r^ = mr2, r^ = mr^, 


V u 

m 

V — u 

If we were to make r^ = mr\, the fifth leg would dup- 
licate the first leg in space relative to the target. To 


116 


THE SEARCH FOR TARGETS IN TRANSIT 


make the fifth leg lie S miles outside the first leg in 
relative space, we must determine so that 


rs + Tj ^ r5 - Va - S 

V u 

from which we find 


rs = mvA + a, 

where (32) 

vS 

a = • 

V — u 


Continuing in this way, we obtain 


7*2 = mr\ 
r3 = mr2 
Va = mrz 
rs = mvA + a 
re = mre 
r^ = mre 


Vg = mr-j 

Vg = mrg — 2a 


rio 

rii 

ri2 

ri3 


mrg 

mrio 


(33) 


mrn 

mri 2 + 3a, etc. 


1. What is the lower limit Ti of T for which the 
search plan for large values of T may be used with- 
out any essential decrease in the probability of de- 
tection for a given amount of searching? 

2. For T less than Ti, what modifications of the 
above plan for large values of T will give an essential 
increase in the probability? 

The curves of Figure 17 of Chapter 1 do not 
answer these questions. It is seen that when t is less 
than (t/u the distribution does not differ much from 
the initial distribution. However, there is a very 
rapid change in the distribution as t increases from 
(7-/a to 2o-/a. For t greater than 2a /u the distribution 
has its maximum at approximately r = ut and moves 
outward at the speed of the target. This transition 
period between an essentially stationary distribution 
and a distribution moving at the speed of the target 
makes the problem difficult. 

The following derivation of equation (31) sug- 
gests a method of handling the problem. In terms of 


A first approximation to n is uT. However, the ap- 
proximation to the spiral by straight legs requires 
that n be slightly less than this value. Equating the 
average distance of the aircraft to the average dis- 
tance of the peak of the distribution from the point 
of fix, the average being with respect to time spent 
by the aircraft on a leg of the plan, we obtain 

n = 0.9uT. 

Since any changes of course of the target will reduce 
the outward component of its velocity and since the 
second circuit is to be flown outside the first circuit, 
we shall reduce n still further and take 

ri = O.SuT. (34) 


Using equation (33) and the obvious relations be- 
tween the lengths of the legs and their distances from 
0, we have 


Li = mvi 
Lg = mLi + Ti 

Lz = mL^ 

La — 'f^Lz ”h a 
Lz = iuLa 
Le = mLz + a 


L^ — ifiLg 

Lg = mLi — 2a 

Lg = mLg 

Lio = mLg — 2a 

L\\ = mL\o 

Li 2 — mL\i 3a, etc. 



the rectangular coordinates {x^y) shown in Figure 12 
the initial distribution is 

If the target were known to travel at speed u in the 
direction d, the distribution at time t would be 


where n and a are given by equations (32) and (34). 

The above search plan has been devised for large 
values of T. For small values of T two questions 
arise : 


F(x,y,t,e) = 


ul cos 0 )^ ut sin 0 )^] 


in other words, distribution (20) with a simple change 
of origin (translation). If, on the other hand, the 


SEARCH ABOUT A POINT OF FIX 


117 


target motion is random in direction, the distribu- 
tion at time t is 

F'(x,y;t,e)dd 

— ^ p- (r2 + uH)f2aK J_ T p ^os (0 - a) 7p 

” 2tct^ 2 Jo ’ 

where y‘^ and tan a = y/x. Using 6 — a 

as a new variable of integration and noting that the 
integral of a function of cos d from — a to 27r — a 
is equal to the integral of the same function from 0 
to 2x, we obtain the distribution given in equation 

(31). 

For a given search plan and an assumed law of de- 
tection, we can apply the above method to find the 
searched distribution at any time t. We first find 
the effect of the searching upon the distribution 
function F{x,y;t,d) and then average with respect to 
6. To approximate this process, choose a small num- 
ber of equally spaced directions 6 from 0 to 2t, the 
number depending upon the degree of approxima- 
tion desired. For each 6 write on coordinate paper 
numbers proportional to the initial distribution and 
think of this distribution as moving in the direction 6. 
Then lay out the search plan relative to this distri- 
bution up to the time at which the distribution is 
to be examined. From the assumed law of detection, 
multiply the distribution numbers by the probability 
that the target will not be detected, taking into ac- 
count the number of legs on which detection may 
occur and the distances from these legs. The result 
will represent the searched distribution for a given d. 
The average distribution then can be obtained by 
averaging the numbers representing the searched 
distributions for each position relative to the point 
of fix and a given direction of reference. 

In applying the above method it was assumed that 
four directions would give sufficient accuracy. A 
check was run by computing a distribution with four 
directions and with eight directions. It was found 
that the two distributions did not differ very much. 
Using a number of values for t and S, search plans 
were laid out and tested as follows: The first circuit 
was decided upon from the curves of Figure 17, 
Chapter 1. Thereafter, each circuit was determined 
by examining the distribution at the end of the 
previous circuit. The following results were ob- 
tained. 

1. If T > (t/w, the original plan for large values of 
T is nearly optimum. 


2. If T < a/u, the plan for large values of T is 
fairly good and may be used if the additional com- 
plication of another plan is not acceptable. However, 
an appreciable improvement can be obtained by 
slight modifications. Lay out the plan as in Figure 
11 with 


Li = mvi 
L 2 = mLi H- n 
L 3 = mL2 
Li = mLz + «! 
L 5 = mLi 
Le = mLz -f Ui 


L7 = mLe 
Z/g = mLj -f- 0,2 
Lg = mLz 
Liq = mLg -(- a2 
Lii = mLio 

Li 2 = mLn -h as, etc. 


and determine n, ai, 02 , etc., as follows: 


n 


S 

m^ -h 1^ 


(35) 


ai = a 2 = • ' ' = ak = vS/v — u, ak + i = — {k 2)ai, 
a* + 2 = + (^ + 3)ai, a* + 3 = —ik + 4)ai, etc., where 
k is the positive integer nearest 2ut/S. Here, n has 
been chosen so that Li and L 3 are separated by a 
distance S. This scheme seemed to be the best pos- 
sible for the first circuit. The succeeding circuits 
then are flown outside this circuit until the point is 
reached at which it is more profitable to search on 
the inside of the distribution. In case T is so small 
that 2uT < #S, a slight improvement can be obtained 
by decreasing ai by 10 or 20 per cent. 

In the case of this section it is assumed that the 
target speed u is known exactly. The question natu- 
rally arises as to how much the probability of de- 
tection is affected by an error in estimating u. 
From the asymptotic approximation to f{r,t) for large 
values of t it is seen that the distribution func- 
tion will be multiplied approximately by the factor 
exp { — FAu^J2(t‘^). This is a very rough estimate of 
the factor by which the probability of detection will 
be multiplied. Thus, if tAu is small compared with a, 
the probability of detection is not affected very much. 

The effect of a speed distribution upon the target 
distribution can be obtained in the usual way. Let 
g{u) be the probability density in speed. Then 

g{u)du = 1 



and the target distribution at time t is 


f 


g{u) • f{r,t)du. 


It is evident from the way in which u is involved in 
}{r,t) that this integral will be difficult to compute for 


2no* f(r,t) 


118 


THE SEARCH FOR TARGETS IN TRANSIT 



Figure 13. Distribution curves for two assumed distri- 
butions of target speed. A: assumed speed u only. B; speed 
of w,0.8m, 1.2m with speed u times aslikelyasO.Swor 1.2m. 

any continuous function g(u) which represents a 
reasonable speed distribution. Consequently, the dis- 


tribution curve was plotted from the curves of 
Figure 17, Chapter 1 for particular values of t using 
the speed distribution function 

g{u) = 0.25 for u = O.Su 
= 0.50 foTu = u 
— 0.25 for u = 1.2u 

where u is the assumed speed. This curve for t = 
5cr/u and the corresponding curve of Figure 17, 
Chapter 1, are shown in Figure 13. The new dis- 
tribution curve has a smaller maximum value than 
the original distribution and is spread out more. 
However, the difference between the two distribu- 
tions is not sufficient to justify any changes in the 
search plans. 


Chapter 8 

SONAR SCREENS 


81 INTRODUCTION 

W HEN A FORMATION of ships, such as a merchant 
convoy or task force in transit, is passing through 
waters in which hostile submarines may be present, 
the danger can be greatly diminished by providing 
for the detection and attack of submerged submarines 
which are closing to its vicinity. With present equip- 
ment, submerged submarines can only be detected at 
sufficient range by hydrophone or echo ranging and 
can be located with sufficient accuracy only by echo 
ranging. This detection must occur as far as possible 
from the convoy, both for its safety against tor- 
pedoes (whose ranges may exceed those of sonar de- 
tection) and to facilitate attacks on detected sub- 
marines. Consequently, it is necessary that the sonar 
equipment be carried on highly maneuverable and 
armed naval units of a relatively expendable nature, 
such as destroyers or destroyer escorts, which are to 
be appropriately stationed at a suitable distance from 
the convoy. Such a disposition of units is called a 
sonar screen. It has the dual function of detection and 
subsequent attack. 

82 TORPEDO DANGER ZONES AND 
HITTING PRORARILITIES 

In designing antisubmarine screens, it is essential 
to determine what the areas are from which the sub- 
marine has a good chance of scoring a torpedo hit 
upon one of the units which it is proposed to screen. 
Speaking loosely, the torpedo danger zone about an 
individual ship or group of ships is the region 
(thought of as moving along with the ships, i.e., it 
is fixed relative to them) within which a torpedo must 
be fired if it is to have any chance of scoring a hit. 
The shape and size of the zone will of course depend 
on the speed and type of the torpedo, as well as the 
speed and disposition of the ships. Speaking more 
precisely, there is a danger zone for each given prob- 
ability P. It is the area from which the torpedo must 
be fired in order to have a chance not less than P 
of scoring a hit. It is bounded by a closed curve con- 
taining the ships, which is the locus of points from 
which the torpedo must be fired in order to have the 


given probability P of hitting. Such curves, one for 
each value of P, are the level lines of the probability 
function p{r,d) and have as equation 

p{r,e) = P. 

The probability function p{r,d) is the value of the 
probability of scoring a hit by a torpedo fired from 
the point of polar coordinates {r,6) with respect to 
the reference point in the formation of screened ships 
(and in space moving along with them). 

It becomes, therefore, our first object to evaluate 
this function p{r,d). The present section is devoted 
to the description of various methods for doing this. 

The primary factor involved in the determination 
of the enemy’s chance of success p{r,B) is the type of 
weapon which he uses. If the lethal coverage of the 
weapon is high, the overall chance of success is cor- 
respondingly great. Consider, for example, three 
types of torpedo. The first runs at a 50-foot depth 
and explodes after a given length of run; the second 
at 5 feet and explodes on contact; the third at 5 feet, 
exploding on contact, but provided with a homing 
device such that whenever it passes within 500 yards 
of a ship it will home onto it and score a hit. It is 
obvious that the first will have a rather small lethal 
coverage, since only a ship in a particular point will 
be affected by the explosion. The coverage of the 
second is greater because any ship along the entire 
run of the torpedo may be hit. The homing feature 
of the third will give it by far the greatest coverage 
because the ships need only be within 500 yards of 
the torpedo track for a hit to be scored. These cases 
can be demonstrated qualitatively by a diagram like 
that of Figure 1. The areas are shown for a torpedo 
proceeding at right angles to the ships. The areas 
shown have the property that any ship whose center 
lies in the area will be hit by the torpedo. For case 
(1) the area covered is slightly larger than the plan 
view of the ship, since a torpedo exploding up to 
about ten yards away might sink the ship. 

Lethal coverage alone does not determine the 
enemy’s chance of a hit. His firing errors must be 
taken into account, and if his errors are so great 
that there is only a small probability that the target 
lie in the lethal area, his chance is correspondingly 
small. For any given accuracy, however, the weapon 


119 


120 


SONAR SCREENS 


with the larger lethal coverage will be the more ef- 
fective. 

The foregoing discussion applies only to a single 
target. When fired at a convoy or group of ships, the 
torpedo will be successful if it hits any member of 
the group, i.e., if any of them lies in its lethal area. 
In general, the torpedo will actually be fired at a par- 

CASE 1 

FIXED EXPLOSION 

n ^sHip 


CASE 2 

CONTACT TORPEDO 


h MAX TORPEDO RUN 


CASE 3 

HOMING TORPEDO 



Figure 1. Lethal coverage for various torpedoes. 


ticular ship, but it may miss that ship and hit another 
one purely by chance. For long-range torpedoes fired 
at large convoys, the chance of such an event may 
be quite considerable. In such a case a torpedo may 
well be fired as a “browning shot” from a fair dis- 
tance, aimed at the convoy as a whole on the chance 
of a random hit. Because of the importance of brown- 
ing shots with large, closely spaced convoys, the 
probability of securing a hit on a merchant convoy 
is quite different from that on a single ship. The 
following discussion will deal with the case of the 
large convoy. The probability of hitting a single ship 
will be discussed in detail (in connection with the 
case of the task force) later in this chapter. 


8 3 the probability of scoring a hit 

ON A CONVOY WITH A SINGLE TORPEDO 

While it has been the general practice to fire tor- 
pedoes in salvo, the probability that a salvo of a 


given number of torpedoes will hit, or sink or damage 
to a given extent, a ship in the convoy, depends on 
the probability of scoring a hit with a single torpedo. 
It depends also, of course, on the assumed law of 
damage produced by a given number of hits (the 
damage being different, e.g., for merchantmen and 
for battleships). The one-torpedo probability be- 
comes, therefore, the fundamental quantity to be 
found as a preliminary to any study of the prob- 
ability of damage to convoys. 

In this section, several methods of estimating this 
probability of a single torpedo hit will be described. 
The data from which such computations start are, 
firstly, the structure of the convoy; secondly, the 
position of the firing point; thirdly, the velocities of 
the units involved. And since the computation must 
be repeated for a large number of firing positions, an 
essential requirement is simplicity and speed of com- 
putation. 

The first method is purely graphical and involves 
no computational difficulties beyond mere counting. 
We draw the convoy to scale with each ship of proper 
size and position. Radiating from the firing point, we 
draw a set of straight torpedo paths in convoy space, 
so chosen as to be spaced in angle with an angular 
density corresponding to the dispersion or error 
which it is intended to assume. If, for instance, 
twenty torpedo paths are to be drawn and the angu- 
lar distribution is considered normal (Gaussian), they 
would be drawn at angles which divide the normal 
curve into equal areas, as shown in Figure 2. 



® ANGLE FROM THAT AIMED AT 

Figure 2. Torpedo firing errors. 


Thus we consider each track drawn as having a 
probability of 1/20 that a torpedo aimed at angle 
= 0 will actually run along it. By superimposing the 
torpedo diagram on the convoy and counting the 
number of torpedo paths that hit ships, we can esti- 
mate the chance of a hit. In the example shown in 
Figure 3, six out of 20 score hits so that the prob- 
ability of a hit from that firing point is 30 per cent. 

In deciding on the probability of a hit from any 
particular point, however, it is necessary to pick out 


PROBABILITY OF HIT ON CONVOY WITH SINGLE TORPEDO 


121 



pick out the best shot by eye after a little experience, 
but sometimes both possibilities must be counted. 

This method is extremely simple and direct, but 
it has a number of disadvantages. In the first place, 
it involves careful drawing and positioning of the 
diagrams. A great deal of inspection of diagrams is 
required. Since only a rather small number of tor- 
pedo paths can be drawn conveniently, the values are 
only accurate to the nearest 5 per cent, and may show 
considerable fluctuations for very small changes in 
firing position or aiming angle. In addition the esti- 
mated probability shows humps and valleys due to 
the screening effect of a regular arrangement of ships. 
From certain angles one ship hides a number behind 
it, while from slightly different ones all are presented 
as targets. The arrangement of ships in any actual 


convoy is not orderly enough to show such an effect, 
so that such variations must be smoothed out more or 
less arbitrarily in arriving at a final result such as 
that shown in Figure 4. 



Figure 4. Typical probability of hit contours. 


122 


SONAR SCREENS 


The choice of torpedo firing error function depends 
on torpedo and predictor characteristics and firing 
doctrines; it will not be discussed here. It should be 
pointed out, however, that this error depends in 
general on the track angle of the torpedo, the angular 
error being small from ahead, large from astern. The 
length of torpedo path in convoy space is also vari- 
able — long from ahead, short from astern. When the 
convoy fired at is large, the variation in firing error 
is not very important and can be neglected quite 
satisfactorily, but the variation in track length must 
be taken into account, as shown in Figure 5. (Figure 



Figure 5. Torpedo track length L in convoy space. 


3 conforms to this requirement.) Since the chance of 
a random hit on a ship not aimed at contributes a 
considerable part to the total hitting probability, 
the variation of the error in aiming is relatively un- 
important. This contribution depends on track length 
but not on aiming error. 

An obvious extension of this simple counting 
method can be used to reduce the fluctuations in 
calculated probability of hitting which are caused by 
the screening effect of a regular lattice of ships. We 
need only consider the possibility of irregularity in 
the convoy formation, that is, of ships being some- 
what out of their proper convoy station. This can be 
accounted for by making each ship a “diffuse target” 
of length equal to the ship’s length, plus the amount 
the ship varies in position (thought of, for simplicity, 
as a fixed amount) relative to the ship’s position as 
aimed at. Then the probability that a torpedo passing 
through the diffuse target will actually hit the ship 
is taken as 


L 



If we now consider the fth torpedo path {i = 1,2, 
• • m; m being 20 in the previous example) and 
the jth diffuse target (j = 1, 2, • • •, n;n being the 
number of ships in convoy), we can define a number 
hij as the possibility that a torpedo traveling along 
the ^th path will pass through the jth diffuse target. 
From a diagram analogous to Figure 3, we evaluate 


j2 


i f 

DIFFUSE TARGET 

Figure 6. ‘‘Diffuse target.” 

the hij as being equal to either 1 or 0 by inspection. 
Then the probability that a torpedo following the fth 
path will hit the jth ship is 

I, 

Pa = K 7 • 

ij 

Hence the probability of one hit on the fth path 
being identical with that of at least one hit on the 
fth path is 

Vi = 1 “ Py)' 

And the overall probability of a hit from the par- 
ticular firing point is 

1 OT r n “I 

i = l 

This probability function is considerably smoother 
and more realistic than that obtained by the simple 
counting method. The diffuse target method is con- 
siderably more laborious, however, and involves a 
good deal of arithmetical calculation. 

It should be pointed out that the choice of the 
lengths of the diffuse targets is rather arbitrary. A 
plausible assumption is that each ship may be out of 
station with respect to its neighbors by a given 



PROBABILITY OF HIT ON CONVOY WITH SINGLE TORPEDO 


123 


amount, which we may call x. Then the length I 
for the nearest neighbors to the ship fired at will be 
L + 2x. If the station-keeping errors are assumed to 
be random, then the I for the next nearest neighbors 
will be L + 2^/2 x, for next I = L 2^/s x, 
etc., until the diffuse targets overlap and the whole 
column becomes essentially one large diffuse target. 




SHIP AIMED AT 


PROPER POSITION FOR NEXT SHIP 


column methods are applicable to straight run tor- 
pedoes without much difficulty. If, however, we have 
a torpedo which enters a convoy and then zigzags 
or circles to increase its chance of a random hit, the 
resulting diagrams become almost hopelessly con- 
fusing. Such we call ‘‘curly” torpedoes. Some calcu- 
lations have been carried out for curly torpedoes on 
a single target by direct methods, but the combina- 
tion of a curly torpedo (or salvo thereof) and a multi- 
plicity of targets is very hard to evaluate in that way. 
It is possible, however, to write explicit expressions 
for the probability of a hit by a torpedo of quite ar- 
bitrary type. While these formal expressions cannot 
readily be evaluated, there are certain simplifica- 
tions possible which lead to useful results and permit 
us to estimate the effectiveness of some types of 
curly torpedoes when used against groups of ships. 


1*^ ACTUAL POSITION OF NEXT SHIP 

Figure 7. Station-keeping error. 

In this picture of the convoy we think of the posi- 
tion of the ship aimed at as being fixed and the posi- 
tions of other ships becoming increasingly uncertain 
the farther removed they are from the ship aimed at. 
It is evident that the actual state of affairs from the 
submarine’s point of view is something of this sort, 
though the specific way in which the uncertainty in- 
creases may vary from that assumed above. If, how- 
ever, the calculations required are prohibitively 
lengthy, a simplification is possible by doing away 
with this variable diffuse picture and treating each 
column of the convoy as a diffuse target in which the 
ships are distributed more or less at random, that 
is, with probability that a torpedo passing through 
it will hit a ship given by 


L 



where a = ship spacing in column. 

This “random column” method is very simple and 
convenient but is not accurate at close ranges. In 
these cases the actual probability of hitting the ship 
aimed at is considerable, and the random column 
discounts this chance. For browning shots fired at a 
considerable distance from the convoy, however, this 
method is very satisfactory. 

The simple counting, diffuse target, and random 


Definitions: 

1. Let Ps{^,y) be the probability density of ships 
at {x,y) (coordinates are in space relative to the con- 
voy throughout) in the sense that Ps(x,y)dxdy is the 
probability that a ship have its center in the in- 
finitesimal area dxdy. 

2. Let (XqYo) be the point from which the sub- 
marine fires its torpedo. It is assumed that only one 
torpedo is fired (see above). 

3. Let f(x,y,Xo,Yo,a) = 0 be the equation of a 
possible torpedo path starting at (Xo,Fo). Variation 
of the parameters a gives the family of all possible 
torpedo paths starting from the firing point. 

4. Let 4>{a,ao) be the probability density that a 
torpedo aimed along path ao actually travels along 
path a. 

The meanings of these symbols are illustrated in 
Figure 8. In such a case the parameter a might be 
taken as the angle between the torpedo path and 
convoy course. The function <l>(a,ao) then describes 
the angular errors which the submarine may be ex- 
pected to make in firing its torpedo. 

In estimating the chance of a hit we will first as- 
sume that the torpedo is known to be traveling along 
a certain path, determined by the parameter a, and 
determine its chance of hitting. This chance is then 
averaged over all values of a, thus allowing for errors 
in firing the torpedo. This process is precisely anal- 
ogous to that carried out for the diffuse target 
method with a straight run torpedo. 

Consider first a short section of the torpedo path. 
Any ship which happens to be close enough to the 
path of the torpedo will be hit, but since we know 


124 


SONAR SCREENS 


only the probability that there will be a ship at any (Xo,Yo) 



Figure 8. Convoy shown as density function. 


ship in to scale, heading on the reciprocal of convoy 
course (corresponding with the fact that the diagram 
is relative to the convoy), with center on the torpedo 
path, and sliding its center along the relative path. 
This is illustrated in Figure 9. This sweep width 
depends on the angle between torpedo path and 
ship’s course and is denoted by S{d). 

For a homing torpedo the effective target area is 
larger than the ship itself and may have a variety of 


forms, depending on the detailed type of torpedo and 
ship involved. To obtain the sweep rate for such a 
torpedo, we follow the same procedure, sliding a 
figure showing the target area rather than just the 
area of the ship. 

As the torpedo proceeds along its path it sweeps 
out the density of ships Ps with sweep width S{d). 
In Figure 10, I gives the distance which the torpedo 
has traveled, and we let p(l) be the probability that 


PROBABILITY OF HIT ON CONVOY WITH SINGLE TORPEDO 


125 



the torpedo will have hit a ship by the time it has 
gone a distance 1. 

In going an additional distance dl it would sweep 
out PsSdl ships (ps and S may be functions of 1), 
but the chance that the torpedo actually reaches the 


In its complete run, then, the torpedo’s chance 
of hitting is 

p(L) = 1 - 

We must remember, however, that this result ap- 
plies only for the particular torpedo path selected 
and that the integration indicated is performed along 
that path with appropriate values for Ps and S. This 
may be indicated by introducing the parameter a 
into the notation as follows, 

p[a] = 1 - ep;-fh"a‘“'. 

Here p[a] is the probability the torpedo will score 
a hit if it goes along the path a. It is then necessary 
to obtain the average value of pla] to obtain the 
probability that a torpedo fired from {Xo,Yq) aimed 
to go on path ao will hit a ship, which we denote by 
P{Xo,Yo,ao). 

P(Xo,Fo,ao) = J (t)(a,ao) pla\da 

all a 

=J <t>(a,a<i) |l - 



distance ‘d” is 1 — pil). Hence the increase in prob- 
ability of hitting is 

dp{l) = p,S [1 - p(l)] dl. 

Solution of this equation gives us 

-log [1 - p{l)] = f p^Sdl, 

Jo 


alia 

It is evident that our previous methods of calcu- 
lation were simplifications of this formula in which 
the quantity in braces was assigned one of a set 
of values by inspection of a diagram, and the func- 
tion </) was approximated by a population of cases 
(in the examples discussed, 20 cases). It is also pos- 
sible to approximate the expression by neglecting the 
error function <f) if we assume that the position of the 
ship aimed at is not definitely known. In this picture 
the torpedo track is taken as perfectly definite, but 



or 


p(l) = 1 - 


Figure 11 . Graphical calculation of probability of 
hitting. 


126 


SONAR SCREENS 


target positions relative to it are indefinite and the 
function is made to absorb the torpedo firing 
errors. Then 

P(Xo,F„,a„) = 1 - ep-/hS“. 

If we make the further simplifying assumption 
that Ps is uniform throughout the convoy and zero 
outside [i.e., Ps = (no. of ships) /(convoy area)], this 
becomes 

P(X„,Ko,a„) = 1 - e 


or -log (1 - P) = PsAo, 

where Aq is the total area swept out in the convoy 
by the torpedo. 

The use of this formula can be illustrated by the 
simple case of a straight run torpedo. We draw the 
torpedo path at a given angle, as in Figure 11, and 
scale off along it distances proportional to l/Ps^S. 
Mark resulting points of division by the corre- 
sponding multiple or submultiple of l/ps*S. Then 
for any position of the convoy shown dotted in Fig- 



Figure 12. Probability diagram for contact torpedo. 


PROBABILITY OF HIT ON CONVOY WITH SINGLE TORPEDO 


127 


TRUE COURSE OF ZIG-ZAG 




ure 11 we can read off the value of —log (1 — P), 
in this case about 0.32. The corresponding value of 
P is about 0.27. In order to carry out a number of 
such calculations for different positions about the 
convoy, we prepare a diagram which has torpedo 
paths at every 10 degrees, similarly scaled off, and 
draw in what might loosely be called isoprobability 
contours. Such a net or coordinate system is shown 
in Figure 12. By placing the convoy (drawn on trans- 
parent paper) over it we can tell at a glance what 
the submarine’s best shot is from any point by pick- 
ing the one along which the convoy has the greatest 
“length” measured in that coordinate system. That 
length also tells us what the submarine’s chance of a 
hit is. One such coordinate system will do for all 
firing positions, torpedo track angles, and sizes and 
shapes of convoy. A new system must be drawn for 
each density, size, and speed of ship in convoy, and 
for each new type of torpedo. 

For curly torpedoes the procedure is more com- 
plicated. If, however, the torpedo is one which has 
a periodic type of motion, such as zigzag or circling, 
an estimate of its effectiveness is not too difficult. 
We can consider the motion of the torpedo (relative 
to the convoy) as made up of a straight-line motion 
along the mean line of advance, with periodic ex- 
cursions superimposed. By plotting in detail the 
course and area swept out for one cycle we can de- 
termine the area swept per unit distance made along 
the mean line of advance. This determines an ef- 
fective sweep width, which we can then use as if 
the torpedo proceeded in a straight course along the 
mean line of advance. Thus we draw the torpedo’s 
mean line of advance and scale off distances in units 
proportional to l/ps*S', where S' is the effective 
sweep width. We must also scale off along it the dis- 
tance that the torpedo actually runs so that its maxi- 
mum range will be properly taken into account. 
These various steps are illustrated in Figure 13. 

Such torpedoes will actually run straight for at 
least part of the run and may then start to circle or 
zigzag as the case may be. The probability diagram 
applying to a straight run is then used out to the 
appropriate distance, and circling or zigzagging dia- 
gram for the remaining yards to run. This is shown 
in Figure 14. 

The chief advantage of this method is that it is 
quick and flexible. Once the required diagrams have 
been worked out for any particular type of torpedo, 
one can readily decide what the submarine’s best 
shot is from a given position, and estimate his chance 


128 


SONAR SCREENS 


FIRING POINT 



Figure 14. Torpedo which runs straight and then 
zigzags. 


of scoring a hit. In a short time this probability 
function can be mapped for all points around the 
convoy. The approximations that are involved should 
be fairly good in most cases. When the range at 
which the torpedo is fired is very short (less than 
1,000 yards), the actual chance of a hit may be 
higher than this method would indicate, since the 
submarine can single out an individual ship and aim 
the torpedo at it with a good chance of hitting that 
particular ship. When the range at which the tor- 
pedo is fired is very long (greater than 5,000 yards), 
the actual chance may be lower because an error 
in firing the torpedo might cause it to miss the 
convoy altogether. In the intermediate range, how- 
ever, the probability calculated in this way should 
be fairly accurate. 

It is possible to reduce the probability at long 
ranges to more reasonable values by introducing a 
very simple <^(a,ao), which gives a rough approxima- 
tion to the actual torpedo firing errors. For instance 
one can take (f) as defined for three angles, the op- 
timum and 10 degrees to either side of it, and equal 
to one-third at each angle. The probability of a 
torpedo hit for each of these three angles can readily 
be averaged to give a rough figure including the 
effect of firing errors. 

Figures 15 to 17 are given as examples of the type 
of results which are obtained by calculations of this 
sort. 




DEEP FORMATION 



0 % 

25 % 

50 % 


50 % 

25 % 

0 % 

TRIANGULAR 

FORMATION 


0 5000 

1 I I I ■ I-J 

SCALE-YOS 

Figure 15. Probability of hit contours for different con- 
voy formations 5,000-yd contact torpedo. Diffuse target 
method of calculation. 



PROBABILITY OF HITTING SINGLE SHIP OF TASK FORGE 


129 



SCALE -YDS 

Figure 17. Probability of hit with 15,000-yd circling 
Figure 16. Probability of hit with 15,000-yd circling torpedo. Random sweep rate method of calculation: 

torpedo. Random sweep method of calculation: no error. “10° error.” 


84 the probability of hitting a 

SINGLE SHIP OF A TASK FORCE 

Ship formations with large spacing between ships, 
as in a task force, allow a relatively simple method 
of determining torpedo probability curves similar to 
those shown in Figures 15 to 17. A submarine firing 
torpedoes at such a formation would presumably fire 
at a specific target, and the increased probability of 
hitting due to neighboring ships would be small. The 
following method, which involves the probability of 
hitting a single ship, can be used: 

o) = — in radians, 
r 

S = Isin 6, 

180 sin 6 . , 

CO = — I m degrees. 

TT 'f 


SHIP 

LENGTH 


FIRING 

I POINT 



Figure 18. Angle subtended by ship. 




130 


SONAR SCREENS 


Assuming a normal distribution of firing errors, let 
a = a {6) be the standard deviation of firing 
errors in degrees at bearing 6 (relative 
space), 


For example, for C = 0.25, m = 0.225, and if I = 250 
yards, 

r = 22,500 


p{r,d) = probability of obtaining a hit from range 
r at bearing 6. 

Q 

p{r,d) = = I 

(T '\/ 27r J —CO /2 

(1) 

2 

= I 

0 - V 27 r Jo 

X 

Setting!/ = 

(7 V 2 

2 ^co/2V2ff 

?(»•,») = ^ I 

V TT Jo 


This reasoning is based on Figure 18 which is 
drawn in space moving with the velocity of the ship 
(the same as the task force) and thus r is the length 
of the relative track of the torpedo. In Figure 19, 
on the other hand, s (which has nothing to do with the 
S of Figure 18) is the length of the torpedo’s track on 
the ocean. This figure shows the relation between 
the various angular errors, i.e., the standard devia- 
tions in angle. The two small lengths marked dx are 
parallel and equal, and perpendicular to the tor- 
pedo’s ocean track s. 

From Figure 19 the law of sines gives the follow- 
ing relationship: 


= erf-4—- 
2V2<r 

Assume that it is desired to find the probability 
contour corresponding to the chance of hitting 
p{r,d) = C. To find the values of r,0 for p{r,d) — C, 
let erf m = C. Therefore, 

CO 

— 7^ = m 

2\/2(t 

180 , sin d 1 
— - I — 7 ^ = m 

TT r 2\^2 (t 

^ _ 180^ sin 0 

27rm\/2 a 


sin d = - sin d 

V 


where d is the the lead angle, d the angle off the bow, 
u the ship speed and v the torpedo speed. It is seen 
that if V is assumed to be known, then the deviation 
of d from the true value will be due to errors in 
estimating u and 6. Let \u and Ad be the errors in 
(2) estimating u and 6 respectively, and let Ad be the 
corresponding error in the value of d as computed 
from the above formula. Then, to a first order of ap- 
proximation. 



PROBABILITY OF HITTING SINGLE SHIP OF TASK FORCE 


131 


A 7/ 7/ 

Id cos d = — sin 6 M cos 6 

V V 

_ -f A0COS0 

^\u 


be applied to determine values of r corresponding to 
the desired probability contour. Following through 
on the example to find the contour for C = 0.25, the 
following table can be computed: 


Since 

cos d = \/f 
we have 

k 

M = - / 

\/l — sin^ 0 

In addition, we should add another term M to ac- 
count for minor equipment and personnel inaccura- 
cies. If the errors Aw, A0, and Ai are independent and 
are distributed normally with standard deviations 
Eu, Ee, and Ei respectively, then the standard devia- 
tion E of the normal distribution of d is given by 
the formula, 

2 

sin^^ + Ee"^ cos^ 0 
1 — sin^ 0 




sin^d = V^l — A^^sin-^, 


sin 0 + A0 cos 0 


(4) 


Since it has been found that E^/u often equals E^, 
the above formula may be written 


E^ = 


k^Ee^ 

1 — k^ sin^ 0 


Ei\ 


(5) 


Before equation (2) can be applied, however, the 
relationship between E and c must be determined. 
From Figure 19, 

dx sin (0 + 90 — 0 = 


sE (sin 0 sin t + cos 0 cos t) = a, 


sin 0 


E 


a sin t 


sin 0 [cos {t — 0)] 

(j sin t 
sin 0 cos d 

sin 0 cos d 
sin (0 + d) ’ 

sin 0 cos d 

sin 0 cos d + cos 0 sin d 

E 


E, 


1 + 


tan d 
tan 0 


( 6 ) 


Using equations (5) and (6) to determine values 
of (7 for various values of 0, equation (2) may then 


30° 


60° 


90° 


120 ° 


150' 


22,500 


sin d 


7.3' 

5.4' 


7.7' 

6.3' 


7.9' 

7.9' 


7.7' 

9.9' 


7.3' 

11 . 6 ' 


2,080 yd 3,100 yd 2,850 yd 1,970 yd 980 yd 


{u = 18 knots, V = 43 knots, Ee = 17°, Ei = 0) 


From the data in the table the curve p(r,0) = 0.25 
can be drawn as in Figure 20. 



This curve represents the locus of points from which 
a submarine firing a 43-knot torpedo can expect a 
25 per cent chance of hitting a ship 250 yards long, 
located at the origin and traveling 18 knots. A tem- 
plate, drawn to the proper scale and having the shape 
of this curve, can be used to determine the prob- 
ability contour about a task force. It must he remem- 
bered that this method does not consider the effect of the 
browning shot. Figure 21 illustrates the use of such 
a template for a task force in a circular disposition. 

The envelope of all the individual curves is the 
desired contour. (This method assumes that the sub- 
marine aims its torpedo at the optimum point, for 
the submarine, in the disposition and that there is a 
ship in the position aimed at.) The placement of 
the screen about the screened unit once the prob- 
ability contours are known is described in the re- 
maining pages of this chapter. 

Only straight run torpedoes are envisaged in this 
treatment. Curly or homing torpedoes against ships 


132 


SONAR SCREENS 



Figure 21. Use of template. 

of a task force would require a combination of con- degrees to starboard (relative), that on the left the 
siderations, those of Section 8.3 and, possibly, de- same relative angle to port, both measured from the 
tailed knowledge of the torpedo characteristics. Such convoy’s course. The included forward region is the 
latter fall outside the scope of the present work. 

8 5 the submerged approach region 

If a submarine is to have the chance C of scoring 
a hit with one torpedo, it must reach a point on the 
curve p{r,d) = C. Since this will be within visual or 
radar range of the convoy, it must make its approach 
to this curve submerged. Let its submerged speed be 
z, the speed of the convoy being u, and assume that,^ 
as in the case of a normal submarine, z < u. Then it 
is not necessarily possible for the submarine to at- 
tain the curve p{r,d) = C. The positions on the ocean 
from which this is possible fill out a region called 
submerged approach region Rc, constructed as fol- 
lows: Draw two tangents to the curve p{r,6) = C, the 
one on the right making an angle oi yj/ — sm~^ (z/u) 

a High submerged speed submarines require separate con- 
sideration, which would be beyond the scope of this work. 



THE PLACING OF THE SCREEN 


133 


submerged approach region (Figure 22). The angle 
and the tangents are the limiting (submerged) ap- 
proach angle and lines, respectively. All this has been 
considered in Section 1.3. 


86 the placing of the screen 

Referring to Figure 22, it is obvious that the task 
which the sonar screen has to accomplish is at most 
to intercept submarines which come from the sub- 
merged approach region Ro (here Ro is drawn cor- 
responding to p(r,d) = 0, the limit of the torpedo 
danger zone) ; for other submarines do not constitute 
an immediate danger, as long as they remain sub- 
merged. Inasmuch as the line efficiency of the screen 
is greater the more escorts there are per unit length, 
the best way of intercepting submarines entering the 
torpedo danger zone from Ro is to dispose them in 
the shortest line connecting the limiting approach 
lines and lying outside the danger zone and inside 
Rq. Such a line is constructed by stretching a string 


around the forward part of the danger zone, its two 
ends being on the limiting approach lines and per- 
pendicular to them {SqSo' in Figure 23). 

Such a screen would give 100 per cent protection 
against submerged submarines if it were perfectly 
tight, i.e., if it had 100 per cent line efficiency. Un- 
fortunately the distances involved require SqSq' to be 
so long that no normally available number of escorts 
can provide a screen with anything like such a high 
line efficiency. The efficiency might, for example, turn 
out to be only 15 per cent, which represents the 
chance of preventing the submarine from approach- 
ing to within easy torpedo range. Then consideration 
must be given to defending less of the torpedo danger 
zone with a shorter, and hence tighter, screen. Sec- 
tion 6.4 discusses the spacing between the screening 
units, and presents a method of determining the 
tightness of a screen having a given spacing. If, for 
example, C = 10 per cent, the level curve pir,d) = 
C = 0.1 being smaller than p{r,d) = 0, a screen along 
ScSc would be shorter and hence tighter than SqSq'. 
If its line efficiency were, for example, 25 per cent. 



Figure 23. Placement of screen. 


134 


SONAR SCREENS 


the submarine would have a less favorable chance 
of hitting the convoy than when the screen SqSo was 
used. If it fires from outside the screen, its chance of 
a hit is, at most, 10 per cent; if it attempts to cross 
the screen so as to be able to fire from a very favor- 
able position, its chance of success is possibly as 
great as 75 per cent. Obviously the submarine would 
do the latter (assuming that it is indifferent to its 
own safety and merely tries to maximize the chance 
of a hit) . But in the previous case this course of action 
would have given an 85 per cent chance of success 
for the submarine. Thus ScSc' would be a better 
screen. This contraction of the screen must, however, 
not be carried too far. If C = 80 per cent, the level 
curve p{r,d) = C = 0.8 might well give a high 
(e.g., 90 per cent) line efficiency. The submarine 
would simply fire from outside the screen and not 
attempt to penetrate it and would have as much as 
an 80 per cent chance of making a hit. What value 
of C must be chosen to give optimum results? 

Assuming always that the only consideration 
governing the submarine’s behavior is the desire to 
make a hit and that it has no primary concern for 
its own safety, then its best course of action is to 
attempt to penetrate the screen ScSc' whenever 1 — 
(line efficiency of ScSc), is greater than C, and will fire 
from just outside the screen if C is greater than 1 — 
(line efficiency of ScSc). In either case its probability 
of scoring a hit (assuming that once it gets through 
the screen undetected it can certainly make a hit) 
is the greater of the two quantities I — (line efficiency 
of ScSc'), C. The situation is visualized by the graph 
of each of these quantities regarded as functions of 
C (Figure 24). The submarine’s chance of hitting is 
evidently represented by the heavy line, i.e., I — 
(line efficiency ScSc') for values of C less than the 
intersection of the two curves, and C itself to the 
right of Co. The optimum screen is the one cor- 
responding with that C which gives the submarine 
the least chance of hitting, ScoSco- This leads to the 
principle : 

To obtain the best screen, use a curve ScSc' of the 
type shown in Figure 23, and bring it in {i.e., increase 
C) until the chance of crossing it undetected just equals 
the chance of scoring a hit from a point just outside it. 

There are several qualifications to be made before 
accepting the above result. 

Firstly, submarines actually do give consideration 
to their own safety; thus, with the screen ScoScf it 
would be more favorable to them to fire from outside 
the screen than to try to cross it. This would continue 


to be true even if 1 — (line efficiency /8c*8c') is somewhat 
greater than C. Hence, from this point of view, the 
“best screen” would be somewhat farther out than 
ScoSco'} just how much farther is a difficult matter to 
estimate. Exactly the same reasoning can be made 
in different words, as follows : If we are going to have 
a certain chance of having one of our ships torpedoed, 
we would prefer to have a greater chance of getting 
the submarine; the best ScSc' should be a little 
farther out than Sc^Scf. In whichever form this 
reasoning is given, it reposes on the fact that a sub- 
marine firing outside the screen is less likely to be 



C 

Figure 24. Optimum tactics diagram. 


attacked than one which tries to cross the screen 
first. 

Secondly, it is not strictly true that once a sub- 
marine has crossed the screen undetected it is sure to 
score a hit. Again this tends to make the “best 
screen” somewhat farther out than ScoScf- If Ihe 
chance of a hit at close range is gf, (0 < gr < I), to carry 
through the preceding reasoning we must replace the 
graph of 1 — (line efficiency ScSc') by their product 
gf[l — (line efficiency >Sc^cO] plotted in the dotted 
curve of Figure 24, and thus reach the best screen 
>Sco'^Co'. 

Thirdly, the submarine may be supposed to have 
more than one torpedo, whereas it has been assumed 
implicitly in the foregoing probabilities that only one 
torpedo was involved. Having n torpedoes would en- 
large the level curves but also increase the damage 




PATROLLING OF STATIONS 


135 


done by a close-range submarine, and these two 
effects would operate in opposite directions. 

Fourthly, it was assumed that a submarine en- 
countering the scr.een does so with equal probability 
over the screen’s entire length. Yet it is markedly 
advantageous to the submarine to operate near the 
sides of the screen so as to facilitate its escape after 
firing. And the normal tracking procedure used by 
submarines tends to bring them into contact nearer 
the front side than the front center of a convoy of 
any size. For both these reasons it is important to 
avoid lessening the screen’s line efficiency near its 
ends, i.e., abeam. 

Fifthly, in the case of screening a fast ship or task 
group which may be zigzagging or maneuvering rad- 
ically, the limiting approach angle is increased and 
the limiting lines are spread out, and hence the 
screen must extend through a greater angle off either 
bow. But the principles developed earlier remain the 
same. 

The extreme case is that of the fast carrier task 
force which must be prepared to change its course 
radically, even to backtrack at short notice (e.g., in 
order to launch planes into the wind, or to avoid or 
surprise the enemy, etc.). Circular screens around the 
whole task force are frequently used in such cases, 
either with equally spaced escorts, or, more effi- 
ciently from the antisubmarine point of view, with 
closer spacing in the forward parts of the circle. 

Even when as radical a measure as a circular distri- 
bution is not necessary, all turning of the convoy into 
unswept waters must be avoided. The screen should 
be extended in the direction the convoy expects to 
turn, so as to detect any submarines possibly present 
therein. This is particularly important in view of the 
tendency of tracking submarines to accumulate along 
the flank : They are surfaced while tracking, but sub- 
merge and become a danger when the convoy turns. 

Finally, it maybe objected that the reasoning upon 
which the choice of ScoSco was based appeared to 
assume that the submarine knew the values of the 
various probabilities involved, a thoroughly unreal- 
istic assumption. Actually this does not invalidate 
the reasoning. We were merely calculating the chance 
of success of the submarine if it did the best thing 
from its point of view. Its ignorance can only result 
in its taking a, for it, less favorable course of action, 
i.e., it will diminish its chance of success. Thus our 
reasoning subsists but does not attempt to figure on 
the chance of the enemy’s making a blunder. To 
figure in such a chance would be to carry the discus- 


sion to a higher order of tactical complexity, not the 
object of the present exposition. 

In order to incorporate the considerations raised 
by these qualifications into the problem of fixing the 
optimum screen, a somewhat less artificial criterion 
of advantage would have to be used. We have con- 
centrated our attention on the 'probability of a hit of 
one submarine with one torpedo, and shown how to 
minimize it. Actually one might want to minimize 
the number of ships sunk per submarine lost (sub- 
marines using salvos of torpedos and many sorts of 
torpedos, etc.), i.e., to posit a more reafistic situation. 
But it does not appear that as far as the present 
general treatment is concerned such a more detailed 
and complicated study would materially alter the 
conclusions. 


87 PATROLLING OF STATIONS 

When the number of escorts is insufficient to pro- 
vide even a moderately tight screen without closing 
to unduly short distances of the convoy, it is cus- 
tomary for the escorts to ‘‘patrol their stations,” that 
is, to take a course which causes their position to 
oscillate about their station sometimes quite a dis- 
tance (e.g., 500 yards) to the right and then to the 
left of the point (fixed relative to the convoy) which 
represents the assigned station. The reasoning upon 
which this process is based is the following. 

When there are too few escorts, the distance be- 
tween two adjacent ones will be such that a sub- 
marine has a very good chance of passing through 
the screen undetected provided it goes about mid- 
way between the escorts; in other words, the screen 
has “holes.” There are two possibilities: Either the 
submarine knows where such holes are or else it does 
not. If it does, it can profit by their presence, and 
thus the strength of the screen will have to be judged 
by its weakest point. If it does not know, it will enter 
the screen at a randomly chosen point, and thus the 
strength of the screen would be measured by its 
average strength, i.e., the average of its probability 
of detection. Now the object of patrolling stations is 
to deprive the submarine of the possibility of utiliz- 
ing the holes, since, when it is near the screen, it is 
proceeding submerged and without being able safely 
to use its periscope. Of course the patrolling must 
have an irregular or random character. Thus patrol- 
ling stations make the second of the above possibili- 
ties the actual one. The average tightness of the 


136 


SONAR SCREENS 


screen is the valid index of effectiveness, and, low 
though it be, it is much higher than the probability 
of detection in the hole. 

Figure 25 shows the situation graphically. The 
ordinate represents the probability of detection at a 
point along the screen represented by the abscissa. 
Patrolling randomizes the situation with regard to 
the submarine, thus replacing the original curve (a) 
by the average ordinate horizontal line (b). The 
area under (a) equals that under (b). 



Figure 25. The effect of patrolling of stations. 


88 PICKETS 

The screen considered here is a line screen. One 
of a different sort and giving an effect of defense in 
depth (e.g., by alternate staggering of ships) would 
have the disadvantage of creating wake interference 
and difficulties of maneuvering. Nevertheless a line 
screen can be given some of the attributes of defense 
in depth by supplementing it with escorts stationed 


2 

I I 




CONVOY 


I I 

3 4 

Figure 26. Stationing of pickets. 

as 'pickets (1, 2, 3, 4), Figure 26. These pickets ac- 
complish the following tasks. 

1. They act as visual or radar observers for sur- 
face submarines; this is particularly important when 
air coverage is lacking, or only present in forward 
sectors. 

2. They investigate suspicious contacts and aid in 
making attacks upon submarines which may be de- 
tected by the screen or by their having attacked a 
ship in the convoy. 



Chapter 9 


AERIAL ESCORT 


9 1 THE TACTICAL SITUATION 

W HEN A FORMATION of ships (such as a merchant 
convoy in transit, a task force, or task group in 
cruising disposition) passes through submarine waters 
(i.e., waters possibly containing hostile submarines) 
safety can be increased by accompanying the forma- 
tion with aircraft (carrier or land based) which per- 
form systematic flights in its vicinity. Such flights 
are the subject of this chapter; they are called the 
aerial escort, the defended ships being termed in all 
cases the convoy. It is to be emphasized that the 
primary object here is defensive — to reduce the danger 
to the convoy — as contrasted with the primarily 
offensive purpose of flights such as the distant beam 
searches made by Aircraft Carrier Escort (CVE) 
killer groups, where the destruction of the maximum 
number of submarines is the primary aim. To put it 
in slightly different terms : If a submarine is present, 
the success of the aerial escort is measured by its 
ability to prevent the submarine from damaging the 
convoy; while the most satisfactory result is un- 
doubtedly the sinking of the submarine, the escort 
must also be regarded as successful if its mere de- 
tection of the submarine permits the convoy to avoid 
it, or even if its presence induces the submarine to 
submerge and remain submerged in a region from 
which no submerged attack upon the convoy can 
be delivered. 

In order to attack a convoy, a submarine must 
first detect the convoy; second, make an approach 
(usually with tracking) to within firing range of the 
convoy; third, fire its weapons; fourth, withdraw to 
safety. Aerial escort is chiefly instrumental in ob- 
structing the first and second of these operations. Its 
first function (prevention of detection) shall be 
called the scouting effect; its second (prevention of 
approach), the screening. 

In any definite situation there will be a maximum 
range R at which a submarine, using all its facilities, 
can detect the convoy. Visual detection of a non- 
smoking convoy in ideal weather occurs at a distance 
limited only by the earth’s curvature and atmos- 
pheric refraction, but it can be greatly lowered by 
adverse meteorological conditions and increased by 
convoy smoking. Radar detection of convoys is alto- 


gether dependent on the set but at present does not 
exceed good daylight visual detection. Hydrophone 
ranges depend on sound conditions and sea state, as 
well as size and speed of convoy; under ordinary 
conditions they are not much in excess of 15 miles, 
but under ideal conditions with a large fast task 
force they may attain 50 miles. R, which is the great- 
est of all these ranges measured from a reference 
point 0 fixed in the convoy and thus taking into ac- 
count convoy size, has in most cases values between 
15 and 30 miles. The circle of radius R and center 
at the reference point 0 is the detection circle. The 
detection circle may be described as the area within 
which submarines have an appreciable chance of de- 
tecting the convoy, and outside of which they have 
little. 



Figure 1, which is drawn relative to the convoy 
(in which, therefore, reference point 0 remains fixed), 
shows the tactically relevant features: The convoy 
and its surrounding torpedo danger zone (shaded), the 
area of submerged approach (see Chapter 8), and the 
detection circle. There is also shown (bounded by 
dotted lines) the area within which the ships of the 
convoy and its escort can detect a surfaced sub- 
marine. The convoy is heading up the page at the 


137 


138 


AERIAL ESCORT 


speed made good of v knots. The region ABCD shall 
be called the danger zone; if a submarine has reached 
it, it is in contact with the convoy and can submerge 
at will and then make a submerged approach un- 
detected by the aircraft; to reach the danger zone is 
a usual tactical objective of attacking submarines. 

The figure is drawn on the assumption that the 
submarine’s submerged speed is less than the con- 
voy’s speed. In the contrary case, the solid curve 
CDAB may have to be drawn much farther back of 
the torpedo danger zone — corresponding to the locus 
within which the endurance of the submarine sub- 
merged just suffices to close the torpedo danger zone. 
Or the curve may be absent entirely, when this en- 
durance is very great. In this case the danger zone 
will be the ring shaped area inside the detection circle 
and outside the dotted curve. 

In order to have scouting effectiveness, i.e., in 
order to prevent the submarine from gaining con- 
tact with the convoy (so that it will either not know 
that the convoy is present, or, in case it has been 
directed to the convoy by means other than its own 
immediate detection, have no precise knowledge of 
the position or course of the latter), it is necessary 
that the aerial escort fly a sufficient amount directly 
outside the detection circle, particularly in the for- 
ward regions, which are those in which submarines 
are most likely to be encountered by the circle 
(Section 1.5). A method of evaluating numerically the 
scouting effectiveness of any given flight plan will be 
set forth in Section 9.2. 

In order to have screening effectiveness, the escort 
flights must be so made that the tracking and ap- 
proach procedures normally carried out by a sur- 
faced submarine are impeded. Essentially this means 
that entrance to the danger zone must be barred, 
and that this area itself must be covered so that even 
if a submarine has entered it and submerged, it will 
have a material chance of being detected should it 
momentarily surface or otherwise show its presence. 
The importance of barring the arc BC is due to the 
fact that a submarine in the path of the convoy but 
unaware of the latter’s presence is likely to be picked 
up by this arc as it moves over the water with the 
convoy, by which time the submarine will detect the 
convoy and then be in a favorable position to sub- 
merge and deliver an attack. The importance of 
barring AB and CD is due to the fact that these will 
normally be crossed by submarines which have 
tracked the convoy on the surface; such submarines 
cannot enter the area of submerged approach nearer 


to the convoy than A or D since they would be de- 
tected by surface units; and an attempt to enter be- 
yond B or C would require their loss of contact with 
the convoy. A precise method of evaluating the 
screening effectiveness of a plan will be found in 
Sections 9.3, 9.4 and 9.5. 

When aerial escort accomplishing these results is 
maintained day and night, maximum protection is 
obtained. Should it be necessary to discontinue the 
flights at night, it is important that predark flights 
of a scouting character be made at greater distances 
around the convoy in order to clear the waters 
through which the convoy will pass in the night as 
well as to detect submarines which may be stalking 
the convoy, intending to close it at night. The char- 
acteristics of such flights will be examined in Section 
9.7. 

The only practicable flight plans for regular aerial 
escort (all those which are flown continuously during 
a length of time, and excluding predark flights) are 
periodic ones. After the period of T hours the plan of 
flight is repeated, the paths flown during the next T 
hours being identical (as seen relative to the convoy) 
to those of the preceding T hours; this continues as 
long as the plan is flown. The period T should be of 
the order of one or two hours. Thus the plot of every 
regular plan relative to the convoy consists of one 
or more closed circuits, the whole being flown every 
T hours. The geographic plot is not closed, but is of 
an advancing recurrent nature, the length of the 
recurrent figure flown by each aircraft being T times 
the aircraft’s ground speed (or average ground speed, 
in case of variations in the latter due to wind or other 
causes) . 

One assumption is presupposed in the quantitative 
probability reasoning concerning any aerial escort 
plan, that the submarine does not know the plan. Other- 
wise any plan using a normally restricted number of 
aircraft would be ineffective. Actually, it is difficult 
to imagine that the enemy could ever obtain suffi- 
ciently detailed information concerning the plan to 
be enabled thereby to penetrate undetected. 

9 2 the scouting EFFECTIVENESS 

A measure of the effectiveness of any aircraft 
escort plan as a method of scouting is the probability 
which it affords of detecting (visually or by radar) 
submarines which are so moving that they will 
eventually enter the detection circle, detecting them, 


THE SCOUTING EFFECTIVENESS 


139 


that is, before they enter the circle.®' It is assumed 
that these submarines are proceeding surfaced (or 
schnorchelling) in a straight course at speed u. The 
probabihty will depend on this speed, as well as on 
the position of the submarine’s path. It should be 
noted that the word “scouting” is used in the present 
study to indicate a primarily defensive operation, in 
contrast to a frequent and more offensive connota- 
tion of the term. 

At this stage it is important to emphasize that 
submarines outside the detection circle are to be re- 
garded as unalerted and either ignorant of the con- 
voy’s existence or insufficiently informed of its posi- 
tion to be enabled to make a systematic approach. 
Consequently they may be regarded as random sub- 
marines; all courses are equally likely. And when at- 
tention is confined to submarines on courses leading 
eventually to their entry into the detection circle, 
they continue to be random submarines but with ap- 
propriately altered distribution of courses. 

For expediency of computation, the probabilities 
of interception before entry into the detection circle 
are evaluated for targets of each velocity class (see 
Section 1.5). In practice this means that, after hav- 
ing estimated the probable cruising speed u of an un- 
alerted surfaced (or schnorchelling, as the case may 
be) submarine, a track angle <f> is selected, and at- 
tention is fixed on the class of surfaced submarines 
of speed u which (1) proceed at the selected track 
angle <f), (2) have tracks relative to the convoy which 
enter the detection circle. Then the probability P^, 
that such submarines be detected by the aircraft is 
calculated. This is essentially a problem of barrier 
patrol evaluation but pertains to the case of barriers 
in which the aircraft flights, though periodic, are 
very irregular. The treatment has been given in Sec- 
tion 1.6, and Chapter 7, and we confine ourselves 
here merely to setting forth a simple method which 
will usually give sufficient accuracy. 

Step 1. Draw the escort plan to scale relative to 
the convoy, and draw the detection circle. 

Step 2. From </>, u, and the convoy speed, find the 
relative course 6 and speed w with respect to the 

a Another measure of effectiveness is the ability of the air- 
craft to force the submarine to submerge outside the detec- 
tion circle, and thus greatly to reduce its chance of detecting 
the convoy. This does not require the aircraft to detect the 
submarine but, rather, the submarine to detect the aircraft. 
Since this depends on the amount of fl3dng in the various areas 
without the circle, in much the same manner as the aircraft’s 
probability of detecting the submarine does, the probability 
of the latter affords a significant norm of evaluation for the 
plan. Therefore this will not be considered further. 


convoy (see Section 1.2). Draw the two tangents to 
the detection circle making the angle 6 with the con- 
voy course. 

Step 3. Measure the total length of that part of 
the aircraft path which lies outside the detection 
circle and between the two tangents drawn in step 
2, and is situated on that side of the circle corre- 
sponding to entrance into the circle by submarines of 
class 

The required probability is then 
= 1 - 

where W = effective search width for the detection 
of surfaced (or schnorchelling) submarines by the air- 
craft (see Section 2.6). 

This calculation of is carried out for a set of 
angles covering the full circle. These may be taken 
equally spaced either in or in 6, at for example 15 
degree intervals (or larger if less complete informa- 
tion suffices). They are conveniently exhibited by 
marking each P^ off radially along a line starting at 
the origin and making an angle 6 — tt (and hence 
directed toward the incoming submarine) with the di- 
rection of the convoy. When the resulting points are 
joined by a smooth curve, a polar diagram (to be 
called the scouting diagram) is obtained. A corre- 
sponding polar diagram of P,^ plotted against 0 or 
<f> — TT can be drawn, but being less directly related 
to the relative picture of the flight plan it gives a 
less clear indication of its scouting tightness. 

It is noted that in the case of fast convoys whose 
speed V is greater than the surfaced (or schnorchel- 
ling) cruising speed u of the submarine, the polar 
curve is a closed loop lying ahead of the polar origin 
0. This corresponds with the fact that along a given 
6 (represented as the radial line at angle 6 — tt with 
the convoy’s course), there are two velocity vector 
angles <^>, one corresponding with submarines pro- 
ceeding toward the convoy, the other to those headed 
away from it but overtaken by it. Thus the loop is 
cut in two points by the above radial line. When 
V < u, there is only one point of intersection, the 
loop enclosing 0. When v = u, 0 is on the loop. 

Actually, the chief use of the scouting diagram is in 
obtaining the scouting coefficient, now to be defined. 
In appraising the scouting effectiveness of a plan, 
consideration must be given to the total detected 
fraction / of all the submarines whose motion will 
lead them into the detection circle. The number of 
submarines of entering the circle per unit time is 
equal to the area 2Rw in relative space which con- 


140 


AERIAL ESCORT 


tains them, times the density of members of C'</, (by 
the randomness, “independent” of </>). Thus their 
number is proportional to w[w = w{(f))]. Of these, a 
number proportional to wP^ is detected on the aver- 
age. Hence, the total number detected is proportional 
to the integral 

1 

2^ Jo '^^'*''>^1'^’^’ 


which is conveniently found by taking the arithmetic 
mean of the numbers w{4>)P<t> found as above for 
equally spaced angles <f). If every submarine were 
detected (P^ = 1), the above would reduce to 


27r 



To obtain the required fraction, the first expression 
is divided by the second, giving 


n2Tr 

I w{(t))P,t,d(f> 

f=-h 

J n2Tr 

I w(4>)d(f) 
Jo 


a weighted mean of the quantities P^. This can be 
computed by taking the equally spaced angles </>!, <f) 2 , 
• • • and obtaining 

r _ 'f^{4>l)P<t>t + w(<t>2)P4>2 + • • • 

^ w{4>i) + w{4>2) + • • • 

or by using previous results [Chapter 1, equation (4)] 
to evaluate the denominator. In the notation of the 
reference, this gives 

, _ arithmetic mean of ic( 0 i)P 0 „k;( 02 )H<^ 2 ,- • • 

^ {u v) (2/T)E{a) 

where v is the convoy’s speed. 

The number /, called the scouting coefficient, is in 
the nature of a figure of merit of the scouting of the 
aerial escort plan.*^ The values are reasonably in- 
sensitive to possible misestimations of u but depend 
very materially upon the value of R. 

Certain modifications of the above procedure may 
be useful in special situations. Thus in the case of 
certain submarines having extra batteries, a high 


b Theoretically, a plan could be designed so that / is maxi- 
mized (subject to the flying restrictions). The direct determina- 
tion of a plan on the basis of this principle seems to be less 
profitable than the indirect method of drawing up a practi- 
cable plan and then checking its scouting value as above, pos- 
sibly altering the plan (without sacrificing other important 
features) for scouting improvement. 


submerged speed may make the submerged over- 
taking of the convoy possible, although only at rela- 
tively short distances, on account of limited battery 
endurance at such speeds. This means that the danger 
zone may be bounded in the rear by a curve passing 
a few miles, e.g., ten, to the rear of the convoy and 
gently concave in its direction, and intersecting the 
detection circle at points near each beam. Then an 
evaluation of the scouting diagram and coefficient, 
with the detection circle replaced by the composite 
curve consisting of the forward part of this circle 
joined to the above rear limiting approach curve, 
may be useful. Such an evaluation proceeds along 
similar lines to the earlier one, except that the dis- 
tance D between the tangents now depends on 0: 
D = rather than D = 2R must be used in the 
denominator of the exponent in the formula for P^. 
This procedure has a certain inaccuracy, inasmuch 
as it counts submarines inside the detection circle but 
behind the rear limiting approach curve as random, 
whereas they can be expected to have gained enough 
knowledge of the convoy’s location to change from 
an accidental to an intended approach. There are 
cases, however, where this inaccuracy may be ig- 
nored. 

93 the screening effectiveness 

A measure of the effectiveness of an aircraft escort 
plan as a screen is the probability which it affords 
of detecting surfaced submarines® which are making 
a systematic approach to the convoy. Such sub- 
marines are not only alerted but they have a fairly 
precise knowledge of the convoy’s location, course, 
and speed. Their approach may include a tracking 
procedure, and some form of closing (e.g., along a 
normal approach course or a collision course) to a 
favorable position for submerging and thence mak- 
ing a submerged close approach and attack. Their 
surfaced approach course may therefore be quite 
complicated, and since the probability of detection 
depends on its geometrical shape, as well as upon the 
phase of the aircraft’s motion (i.e., its position on 
its circuit when the submarine is at a given distance 
from 0), the evaluation of all such probabilities is 
not feasible. Hence the importance of the following 
line of argument, intended to establish as a useful 
norm of screening effectiveness the probabilities of de- 
tection calculated for collision courses only. With the 

cOr submarines which by schnorchelling or using their peri- 
scope are affording the possibility of aerial detection. 


THE SCREENING EFFECTIVENESS 


141 


reference point 0 in the convoy as center, a circle is 
drawn so large that it includes the whole aircraft 
escort track so far within it that there is no prob- 
abihty that any of the aircraft detect a submarine 
outside it. This circle, in general much greater than 
the detection circle, will be called the reference circle. 
Any approaching submarine will cross the reference 
circle at some (last) point A at the bearing 6 relative 
to the convoy’s course. This is shown in Figure 2, 



Figure 2. The reference figures in an approach. 


drawn relative to the convoy. One characteristic of 
the submarine’s path is d; another is the distance from 
the convoy at which it submerges. Figure 2 shows 
three paths, AB (the collision approach), ABi, and 
AB 2 , all corresponding to the same 6 and same dis- 
tance from convoy of point of submergence (thus B, 
Bi, 5 2 are all on the same “submergence circle”).^ 
How do such paths compare in their probabilities 
of detection? Again it must be emphasized that the 
submarine is assumed not to know the aerial escort 
plan : the most it knows is that aerial escort is being 
flown. Otherwise aerial escort with any normal num- 
ber of aircraft could not be assumed to be useful, 
inasmuch as the submarine could be presumed to 

dWe are assuming that the boundary of the submergence 
region is a circle, thus considering only the simplest case. With 
a tactical situation known in more detail, some other region 
may be more appropriate. For example, the danger zone has 
been used in analyzing the plan of Figure 17, but the principles 
of the reasoning remain the same. 


know how to get in undetected. Consequently, the 
paths AB, ABi, etc. are randomly situated with re- 
spect to the aircraft paths. Now of all the submarine 
paths AB, ABi, etc., the collision course AB is the 
one which takes the least time for the submarine to get 
from A to the submergence circle. If therefore, the 
density of air coverage is substantially the same 
throughout the regions through which AB, ABi, etc. 
can be expected to pass, the probability of detection 
for AB is less than that for ABi, AB 2 , etc. On ac- 
count of the intention of tracking the convoy before 
closing, and then of entering the submerged approach 
region (Figure 1), many submarine paths may be 
expected to be of the type of ABi, hence, if the prob- 
ability of detection of A B is to represent a genuine 
conservative estimate (lower limit) of the probabili- 
ties of detection of ABi, AB 2 , etc., the flight plan 
must provide adequate barring of crossing the line AB 
of Figure 1. 

Consider now two collision courses: AB and AO 
(Figure 2). The probability of detection of AB is 
obviously less than that of AO (at most equal to AO, 
in the case when all flying is done well outside the 
submergence circle). If the density of flight were es- 
sentially the same at all the places through which 
the collision course of bearing angle d passes, out to 
a point D, and then fell to zero, the probability of 
detection of A 5 would be given by multiplying that 
for AO by the factor BD/OD. If there were more 
flying farther out along OD than at points nearer to 
0, the factor would be greater than BD/OD; if less 
flying far than near, the factor would be less than 
BD/OD, and, indeed, the probability of detection of 
AO gives little information about that oi AB when 
the flying is highly concentrated near the convoy; but 
such plans are bad plans (since they duplicate the 
sighting operations of the surface escorts and since 
they cover regions where the submarine can be ex- 
pected to be already submerged) and need not con- 
cern us here. Their exclusion is part of the task of 
the design of a plan (Section 9.6 below). Thus, with 
reasonable escort plans, the probability of detection 
of AO is a satisfactory index of the tightness of the 
screen along the line of bearing 0. We may add that 
the probability for AO is the same (on account of 
the way in which the reference circle was drawn) 
as that for a collision path of bearing 6 but extend- 
ing to infinity instead of to A. 

In conclusion it is here posited : 

Under the proviso that (1) the entrance to the 
submerged approach region be adequately guarded 


142 


AERIAL ESCORT 


(so that probability of detection of ABi is not less 
than AB), and (2) the flying be not unduly con- 
centrated near the convoy, the probability piS) of 
detection of a collision course submarine coming from 
infinity to 0 along the bearing 6 calculated for various 
representative values of 0 is a norm of the screening 
effectiveness. The polar diagram r = p{d) is called 
the screening polar diagram. Its construction is the 
subject of Section 9.4. 

A modified procedure is to take the polar diagram 
in which probabilities of detecting collision course 
submarines outside the danger zone are plotted, in- 
stead of probabilities of detection all the way in to 
0. This occasionally gives useful indications, but its 
value is mainly in scrutinizing poor plans, i.e., those 
in which an undue amount of flying is done close in. 
In a well-designed plan, the diagram as it has been 
considered earher should not be too different from 
the modified case here mentioned. 


9 4 the screening polar diagram- 
general CONSIDERATIONS 

As in Section 9.2, the period T of the flight plan is 
ascertained, and the diagram of the plan is drawn in 
convoy space. The problem is to find the probability 
p{d) of detecting a surfaced submarine approaching 
on the collision course which in convoy space is the 
radial line drawn out from 0 at the angle 6 with the 
convoy course. In order to see precisely what is in- 
volved, we shall give the exact formulation of p{6), 
and afterwards study methods for its approximate 
evaluation. 

Let t denote the time (hours) before a particular 
submarine reaches 0 (at 0, t = 0). Marking an x 
axis in the direction of the convoy’s heading (the 
reference point 0 in the convoy being the origin) 
and a y axis in the starboard direction (Figure 3), 
the submarine’s position at any time t is and 

if w{6) is its relative speed, we have 

Xt = w{6)t cos 6 
yt = w{d)t sin B, 

which are, in fact, its equations of motion. 

If {Xt\ Yt^) are the coordinates of the fth aircraft 
at time t, this aircraft’s equations of motion are of 
the form 

= X\t), 

Y,^ = Y\t\ 


where the two functions X\t) and Y\t) are each 
periodic and have T as a period. Indeed, T is the 
smallest common period of all the functions X^(t), 
Y\t) for all values of i (i.e., the least common multiple 
of the periods of all aircraft circuits of the plan). Of 
course some of the aircraft may repeat their circuit 
more often than others, so smaller numbers, such as 
T'/2, may be periods of some of the pairs of functions 
X\t), Y\t). It must be realized that to give the 
escort plan is to give these functions, and vice versa. 

Let 

7(V(x - xy +(y- YY)dt 


be the instantaneous probability of sighting a target 
(surfaced submarine) at {x,y) by an observer (air- 
craft) at (X,F) (see Section 2.2).® As in Chapter 2, 



Figure 3. Collision approach: the coordinate system. 


the probability of sighting the submarine on its entire 
collision course by the fth aircraft is 

1 - exp I - 7 1^\/ (x, - Xt^y + {yi - 

and the probability that at least one of the aircraft 
sight it is 

1 - exp|- 


e Effects of target aspect and bearing on this probability are 
being ignored in thus assuming that 7 is a function of the 
distance between observer and target. 


SCREENING POLAR DIAGRAM — GENERAL CONSIDERATIONS 


143 


the summation in the exponential being over all 
values of ^ which occur (e.g., from 1 to 3 if there are 
three aircraft flying the plan). 

This is not yet the value of p(0). 

Suppose that h second submarine is following the 
first at a distance w{d)T behind it (i.e., r hours later). 
If r = T, 2T, ST, etc., its chance of detection is the 
same as before, because of the periodicity of the 
plan. But if 0 < r < T, it will arrive at the various 
points of its path when the aircraft are at a different 
phase in their circuits from those of the previous 
case, and thus the probability of detection will be 
different. To find its value, we have but to write the 


The exact computation of p{6) would be a formid- 
able task and is not warranted, in view of the fact 
that any explicit expressions for y and w{d) are at 
best only very approximate, while at the same time 
there is bound to be considerable navigational in- 
accuracy in flying the plan. Fortunately, approxi- 
mate methods exist whereby p{d) can be evaluated. 
The earliest method was to use a definite range law 
of sighting and treat the problem graphically. This 
method, except as giving very crude indications, has 
had to be abandoned. For it not only gives an un- 
satisfactory degree of approximation, but, more im- 
portant, it frequently yields a value p{d) = 1, thus 



CONVOY 

HEADING 



A 


Figure 4. Types of paths crossed. 


B 


equations of motion of the second submarine {xt,yt), 

xl = + r) COS0 = 4.^, 

yl = w{Q){i + r) cos d = 

and substitute these expressions into the above 
formula in place of (xt,yt), i.e., replace {xt,yt) by 
{xt + r) Vt + t)‘ Now the value of p{d) is the prob- 
ability of detecting a submarine chosen at random 
on the line of bearing 6. Its position is characterized 
by the value of r, which is uniformly distributed in 
the interval 0 ^ r < T. Thus to obtain p(d) we 
must average the prohahilities obtained as before for 
each r; we have then finally 

= a 

rt = Vixt + r - +{yt+r- Y,V . 


giving the impression that a perfect barrier could be 
established (corresponding with the idea of “sweep- 
ing the ocean about the convoy clean”). This is an 
illusion: Every plan gives the submarine a finite 
(though possibly small) chance of an undetected sur- 
faced passage through the aircraft screen, provided 
the velocities make such an approach kinematically 
possible. A much better method, for the visual case 
at least, is the one based on the inverse cube law of 
sighting. It is set forth in detail with the aid of 
examples in Section 9.5. Here we merely outline the 
general principles of the approximation to p{d), what- 
ever be the nature of 7, and thus applying equally 
to any law of visual or of radar detection. 

As the first step in the approximation, suppose 
that the submarine crosses^ two partial paths Ci, 

H.e., either actually intersect, or pass in close enough prox- 
imity to afford an appreciable chance of a sighting. 


144 


AERIAL ESCORT 


C 2 , of the plan (and possibly others). This may occur, 
as in Figure 4 A or 4B. 

The distinction is that in (A) the paths are either 
flown by different planes or by the same plane at con- 
siderably different epochs, and therefore, because of 
the irregularities in the flights and in the submarine’s 
motion, etc., the detection hy the aircraft on Ci and 
the detection on C 2 may he regarded as statistically 
independent events] whereas in the case (B) such in- 
dependence may not he assumed. Furthermore, de- 
tection on any of these paths will not occur (i.e., 
occur with but a negligible probability) at a con- 
siderable distance from the submarine’s path, for 
example on the dotted portions of Ci and C 2 and 
beyond. Thus we may at our convenience either sup- 
press all this more distant part of the paths or al- 
ternatively (when it is mathematically simpler) pro- 
duce the straight lines to infinity away from the sub- 
marine’s course. It then becomes a much simpler 
problem to calculate the probability of detection by 
Cl and then by C 2 separately in (A), and by the 
combination Ci + C 2 in (B). And we are thus led to 
the following first simplification in the computation 
of viff)- 

For each given B, separate the aircraft’s paths into 
coherent pieces [Ci and C 2 separately in (A), Ci + C 2 
joined in (B)], and regard each piece simplified at its 
distant parts either by suppressing them or producing 
them to infinity as straight lines. Then compute the 
probability of detection for each part by the methods 
to be developed below. Lastly, combine the probabili- 
ties of the different parts as independent probabilities. 
Thus, in (A), if pi{e) and p 2 {B) are the probabilities of 
detection by Ci and C 2 , and if no other paths are 
crossed, we have 

p{e) = 1 - [1 - pm [1 - pm> 

The second step in the approximate computation 
of p{B) concerns the evaluation of the probability of 
detection p'{B) for a single coherent part of track 
W = Vi or P 2 in the previous example). Figure 5 
shows the situation schematically. The coherent part 
of track is C (which may also be thought of as bent 
back, as in Figure 4B, where C = Ci + C 2 ). 

Since the aircraft flying C has a much greater speed 
than the submarine, the latter will move only very 
slightly during the time it takes the aircraft to 
traverse that part of C (the solid line) which is close 
enough to the submarine’s path to give any ap- 
preciable chance of detection. Hence the approxi- 


mation: Regard the submarine as stationary [e.g., at 
the point {x,y)] while the aircraft makes a given flight 
of C. The formula considered earlier then gives for 
the probability of detection during this particular 
flight 

1 - exp Jt [v^(X(<)-x)2 + (7(0-2/)'']*} 

- 1 - { -£, [^(.v(!) - .)■+ (l- (?) - ,)’]f } 

= 1 - exp|-J^/(r)ds 

where v is the aircraft speed, s is the distance 
along its path C measured from any fixed reference 



Figure 5. Successive positions seen during approach. 


point, and ^ denotes integration over the whole of 

C. J' f(r)ds is the contact potential (Section 2.3), r 

being the distance between the target at the fixed 
point (x,y) and the observer (aircraft) at the moving 
point (X,Y). A further approximation is involved 
here, in that the average ground speed v is taken as 
the aircraft speed in convoy space. Now C is flown 
regularly every T' hours, where T' = T or an in- 
tegral submultiple thereof (e.g., T' = T/2). Hence if 
at one epoch (time when the aircraft is at a fixed 
reference point on its path C such as A, Figure 5) 
the submarine is at ix,y), T' hours earlier it was at 


SCREENING POLAR DIAGRAM — GENERAL CONSIDERATIONS 


145 


(x',2/'), and T' hours later it will be at where 

x' — X = X — x’' = T'w(d) cos 6, 

y' - y - y - y" = T'w{e) sind; 

and there will be other earlier and later positions of 
the submarine corresponding with multiples of T'. 
The probability of detection in at least one of these 
positions is 

1 — exp I — 

where 

r„ = V{X-x„y + {Y-y„y, 

{^n,yn) being the nth position of the submarine, e.g., 
lx„,yn) = (x,y) {x’,y’), {x",y"), etc., and X = X(s/v), 
Y = Y{s/v)j the coordinates of the (moving) air- 
craft. As before, this must be averaged over a com- 
plete set of representative positions of {x,y), e.g., over 
all positions specified by 


struction of the segment B'B of the submarine’s 
path, where B'B is the locus of points not farther 
than D from C (Figure 5). It may happen that 
B' = 0; but in all cases the notation is such that B' 
is nearer to 0 than is B. Figure 6 shows a scale at- 
tached to the submarine’s path : B' has the coordinate 
zero, B has h (b — B'B), a typical position of the 
submarine is etc. And on this scale the points wT', 
2wT', etc., have been marked [w = w{d)]. Figure 6 
(A), (B), and (C) show three typical cases. 

In case (A) there is at most one opportunity for 
an aircraft flying C every T' hours to detect the sub- 
marine. With the probability h/wT', the submarine 
will be on B'B for some flight of C, and having moved 
at least wT' miles between flights, it will not be on 
B'B at any other flights of C. Hence we have 

= {l-exp[-X/(r,)d«]}di, 


[x + Tw{d) COS d,y Tw{d) sin 0], 0 ^ r < T'. 
Thus 

[“^»X 

The third step in the approximate evaluation of 
p(6) aims at simplifying the summation in this 
formula for p'(0). To this end, select a distance D 
having the following property. At a lateral range 
greater than D, an aircraft on a straight course has 
a negligible chance of sighting the submarine, where- 
as at less than D it begins to have a chance of sighting 
which must be taken into account. Thus, ii D = 2E 
{E = effective visibility), the chance of sighting at 
lateral range D is of the order of 2 per cent; and for 
definiteness, this value of D shall be used herein. The 
decision to neglect points of the submarine’s path 
farther from C than the distance D leads to the con- 


where 


= 





is the distance between the aircraft at the point on 
its path corresponding with s, and the submarine’s 
position {x^,y^) corresponding with Time averag- 
ing used previously has been replaced by the equiv- 
alent process of length averaging, and all terms but 
one in the summation are absent. 

In case (B), there will surely be at least one op- 
portunity for the aircraft to sight the submarine, and 
surely not more than two such. The case of just one 
occurs when the submarine’s position ^ is in the inter- 
val {b,wT'), an event of probability (2wT' — b)/wT' . 
If this occurs, the chance of detection is 




B' ^ B 

0 ' b 

B' 

0 b-wT' wT' 

B' ^ 

w'r 


w'T' 

B 

b 


y{k)b ^ wT' 

-^^HB)wT' < b ^ 2wT' 
B 


0 


2wT' 
Figure 6. 


6 


-y{C)2wT' < b 


146 


AERIAL ESCORT 


The case of just two opportunities occurs when the 
(one and only one) ^ in (0,T'w) falls in the interval 
(0,6 — wT'), an event of probability (h — wT')/wT'. 
Then the second and last opportunity occurs when 
the submarine is at ^ + wT\ which will necessarily 
be in {wT',h). The chance of detection in this case is 

r^i {i-exp[-X{/(n) 

+ /(?•{ +«.r)| dsjj. di. 

When these expressions are multiplied by 
(2wT'— h)/wT' and (6 — wT')/wT' respectively, and 
the products added (corresponding to total proba- 
bility of two mutually exclusive events) the final 
expression for p'{d) is obtained. 

This assumes an absolute accuracy and regularity 
of flights and submarine motion lasting through the 
very appreciable time T', an assumption never real- 
ized in practice. It is more realistic, as well as mathe- 
matically simpler, to regard the two opportunities of 
detection as independent events and to combine them 
as such. If their individual probabilities are, re- 
spectively, Pi and p 2 , so that 


Vi 


1 

h-wT 



exp 




]} 


dt 


Pi 


b=W'L' 


then their combination is pi P 2 — P1P2. 

In case (C), there are many positions of the sub- 
marine, differing successively by wT', at which the 
aircraft may be in sighting range. When C (Figure 5) 
is not close to 0, it is usually sufficiently accurate to 
regard the summation 2,^ as an infinite sum, i.e., to 
employ formulas for infinitely many congruent and 
equally distant sweeps (in target space). When C 
is close to 0, this method must be modified to take 
account of the fact that the sweeps occur essentially 
on one side of the target (in target space), e.g., the 
probability p given by the sweep formula is replaced 
by 1 — Vl — p to obtain p'{d). Intermediate cases 
would offer greater complications in a direct treat- 
ment, but in practice it is usually sufficient to com- 
pute the two extreme values cited above and accept 


an intermediate value, such as their arithmetic mean. 

It does not appear profitable to carry the discus- 
sion further in general terms, i.e., without assuming 
a definite form for 7 or/(r). 


9 5 PLOTTING THE SCREENING POLAR 
DIAGRAM: INVERSE CURE LAW 

The practical problem of plotting the screening 
polar diagram when the escort plan and various 
relevant quantities are given is a task of approximate 
numerical computation of the probabilities p{d) dis- 
cussed theoretically in the last section. It can be 
carried out only when the law of detection is given, 
either as an equation, or a table or graph of the 
function /(r). Chapters 4 and 5 have studied the law 
in detail in the cases of vision and radar and repre- 
sent the best information existing at the time of 
publication. Their results are mainly in tabular or 
graphical form, and their application to the present 
problem involves a rather lengthy set of numerical 
computations. Partly to illustrate the principles of 
procedure in as simple a form as possible and partly 
to show how the computation was actually done in 
the case of many plans which have become fleet 
doctrine and had to be designed before the more de- 
tailed information of Chapters 4 and 5 was available, 
the present section will be based on the inverse cube 
law of detection (see Section 2.2). But it must be 
borne in mind that this is at best approximate and 
is not applicable in the case of any considerable 
amount of haze (low meteorological visibility). How- 
ever, it is doubtful if when this law shows that a 
certain plan is better than another a more accurate 
law would reverse the comparison. 

The inverse cube law finds its expression in the 
equations 

f{r)ds = ^, m = 0M6E^, (1) 

where r is the range and E the effective visibility in 
nautical miles (see Chapter 2, equations (10) and 
(46), with dt replaced by ds = dt/v in the former. 
From these, the process of Section 2.3 of Chapter 2 
as illustrated by Figure 5 of that chapter leads to 
equation (23), as shown therein. It is expedient to 
rewrite these results in a slightly different form, giv- 
ing Figure 7 below, which is drawn relative to the 
target (rather than the observer, as Figure 5 of 
Chapter 2). In other words, the target is fixed in 


PLOTTING THE SCREENING POLAR DIAGRAM 


147 


Figure 7 , and the observer (aircraft) is flying with 
the relative velocity w. The angles co', co" of the earlier 
treatment are replaced by 7', 7" (not to be confused 
with the instantaneous sighting probability density). 
Thus we have * 


p = 1 - 


(2) 

dy r 

= m I 
r* Jy 

y" dy 

' {x'^ + y^Y 

(3) 



It must be remembered that x in the above equations 
is the distance along the normal to the aircraft path 
from the target. Of equations ( 4 ), ( 5 ), and (6), only 
equation (6) is simple enough to be put through the 
averaging process which is necessary when the sub- 
marine position is not fixed. Thus, it will be found 
necessary to substitute infinite aircraft tracks for the 
finite ones found in the actual escort plan to be 
evaluated. 

Often the aircraft will fly a path such as is in- 
dicated in Figure 8 A and B. This path is made up 
of two semi-infinite straight lines joined at the turn- 



Figure 7 . Detection at fixed speed and course, drawn relative to the target. 


The quantities r' and r" are the ranges from the 
respective limits of the aircraft track, y' and y", to 
the target. These extreme range lines make the angles 
7' and 7" with the aircraft track. Thus, substituting 
in ( 3 ): 

^ = ^2 (cos t" - cos 7')- (4) 

Should y' go to — 00 or y" to + 00 ^ the above equa- 
tion, substituted in (2) above, becomes 

p(x) = 1 - ( 5 ) 


ing point 0 . Using equation ( 5 ), since the contact 
potentials are additive (see Section 2 . 2 ), it is possible 
to derive a convenient form for the combined con- 
tact potential. 




+ 


(1 + COS 72), ( 7 ) 


{x sin 72)^ 
ml/ .,71, o T2\ 

x 2 2 V 2 2 ) 


where 7 is the appropriate one of 7', 7". Similarly, 
for a flight from — 00 to + <» , 

V{x) = 1 - (6) 


By setting m' = 0 . 046 (£/')^ 


= 0 . 046 ('^^" 


csc^ — + csc^ 



148 


AERIAL ESCORT 


the form of equation (6) may be obtained : 

Vix) = 1 - (9) 

Target known to he on a straight-line interval. If, in- 
stead of a stationary target, we have a moving one, 




B 

Figure 8. Submarine locus passing through angle of 
aircraft path. 

the above formulas can still be applied to solve the 
problem. It is first necessary to consider cases where 
the submarine position at the times of the aircraft 
pass is known to be in a given straight-line interval 
and equally likely at any point therein. The case in 
which the aircraft flies along a straight infinite path 
which meets the submarine path at right angles at 0 
is shown in Figure 9. The submarine is taken to be 
‘ 'uniformly distributed” in the interval from 0 to A 
(length a). Since the chance that the target be be- 


tween X and X dx is dx/a, the probability of de- 
tecting the submarine is : 

pia) = — f (1 — dx, (10) 

ajQ 

In the more general case where the angle a in 
Figure 9 may have any value, it is necessary to use 
as lateral range x sin a. The following equation is 
obtained. 

via) = - f (1 - (U) 

a Jo 



Figure 9. Straight aircraft path: averaging process, (a =OA.) 


Similarly, the case of the bent aircraft path can 
be expressed in this form. Figure 10 illustrates the 
situation. (The submarine may be at any place be- 
tween 0 and A.) The expression for the probability 
involves the quantity m' defined in equation (8) : 

p(a) = i f (1 - dx. (12) 

a Jo 

Equations (10), (11), and (12) can all be inte- 
grated to yield 

p(a) = = 1 - e-x’ + (1 - erf X), (13) 

where 

X = 0.303 — • (14) 

a 

For straight aircraft paths (Figure 9), 

E' = E CSC a) (15) 

for bent aircraft paths (Figure 10), 

£' = |.Jcsc^|* + csc>i|^- (16) 


PLOTTING THE SCREENING POLAR DIAGRAM 


149 



A 

Figure 10. Aircraft on a bent path. 


The principal curve on Plate I plots g{E'/a) against 
E'/a (for this curve the abscissa E'/wT, is equal to 
E'/a). 

The formulas heretofore derived can be applied to 
a more general case, namely, that in which the sub- 
marine is equally likely to be at any position in any 
straight line interval J5 to C; if the aircraft turns, its 
turning point must lie either on BC or on BC ex- 
tended. The first case to be considered is that in 
which B and C are on the same side of the aircraft 
path, as shown in Figure 11. 

For purposes of derivation, let us assume that 
there is a submarine in the interval 0 to B and that 
it is equally likely to be at any position in the in- 
terval. The probability of detecting that submarine 


is Pfi, and can be calculated with the aid of equation 
(13). It can also be calculated as shown below in 
terms of pc and yBC, which are defined as follows: 

Pc = the probability of detecting the submarine if 
it is given in the interval OC. 

Pbc = the probability of detecting the submarine if 
it is given in the interval BC. 

[pc, as well as Pb, can be calculated by equation 
(13).] _ 

c/h = the probability that the submarine is in OC. 
d/h = the probability that the submarine is in BC. 

Thus, with the submarine anywhere between 0 
and B, the product c/h{pc) is the chance of detecting 
the submarine between 0 and C, while d/b(pBc) is 



Figure 11. General disposition of target with respect to aircraft path. 


150 


AERIAL ESCORT 



Plate I. Curves for the computation of detection probabilities. 



Figure 12. Case in which target can be on either side of aircraft path. 



PLOTTING THE SCREENING POLAR DIAGRAM 


151 



Plate IL Curves for combining independent probabilities. 


the chance of detecting it between B and C. Adding 
the two together gives 

Pb = pc ^ PBC, (17) 

and, transposing, we obtain the desired probability, 
h c 

PBC = ^ ^ PC'* (IS) 

The second case, illustrated in Figure 12, occurs 

when 0 falls inside the interval BC. Following the 
reasoning of the previous case, the obvious conclu- 
sion is: 


c b 

PBC = ^pc + ^ PB, (19) 

where both pc and ps are calculated with the aid 
of equation (13). When the aircraft’s path is not 
straight, as in Figure 12B, the correct angles must 
be used to calculate pc and While 71 and 72 apply 
to Pb, their supplements are used in computing pc. 
Thus for Pc equation (20) becomes: 

r . + 

= f ^ <20) 


152 


AERIAL ESCORT 


The aircraft, in some situations, will fly parallel 
or nearly parallel to the locus of possible submarine 
positions, as in Figure 13. Referring to equation (6) 
for an infinite aircraft flight past a stationary target, 
it is seen that the probability of detection at any 


B 


LOCUS OF 
SUB POSITIONS 


Figure 13. Case of parallel aircraft path. 


one submarine position, and, therefore, along the 
entire locus, is given by the expression: 

Pparalle. = 1 " « (21) 
^ I _ g-0.092(£?/x)2 

A plot of equation (21) versus the quantity E/x is 
shown in Figure 14. If the submarine is assumed to 
be in the interval BC and its course is parallel to, or 
makes a very small angle with, the aircraft’s course. 



012345678 

E/x 

Figure 14. Probability of detection in parallel flight. 


the probability of its detection on a single infinite 
flight is given by the ordinate corresponding to the 
appropriate value of E/x. For nearly parallel courses 
the average value of x may be used. 

Target moving on a straight path. Let us reconsider 
the foregoing cases (Figures 11 and 12), assuming now 
that the plane repeats its “infinite” path every T 
hours (the period of the actual screening plan). It 
is desired to find the probability of contacting a sub- 
marine while it is moving from point B to point C 
with the velocity w. This w is small enough compared 
with the aircraft speed so that the submarine motion 
during a particular aircraft pass is negligible. Be- 
tween passes the submarine covers the distance wT. 
The time at which the submarine reaches point B is 
completely at random. However, the situation re- 
peats itself every T hours. A segment of length 
wT may be chosen arbitrarily on the submarine’s 
(straight) path. The submarine will be somewhere 
in this wT section during one of the aircraft passes. 
During the preceding and succeeding passes the sub- 
marine will be in the adjacent wT -length segments. 
This set of nonoverlapping wT segments should be 
extended to include at least all of BC. Since the 
rigorous solution of the problem is independent of 
the position of the set ot wT segments relative to BC, 
we shall in each case choose a set for which the errors 
due to various approximations will be minimized. 

If d, the distance between B and C, is less than or 
equal to wT, the submarine is exposed to only one 
pass, for it will be outside BC at the time of any other 
pass. Choosing Si wT section which encloses BC, the 
submarine is equally likely to be anywhere in this 
section at the time of the associated aircraft pass. 
d/wT is the chance that the submarine is between 
B and C and thus exposed to the detection. The earlier 
paragraphs have given the chance {vbc) of contact- 
ing a submarine which is definitely in BC. There- 
fore, the desired probability (p) is (d/wT)pBc- Then 
when d < wT, equations (18) and (19) become, re- 
spectively : 



d 

P — ^yiPparallel, 


(24) 




PLOTTING THE SCREENING POLAR DIAGRAM 


153 


where, in each case, 



In equation (25) above, a is the distance along the 
submarine path from the aircraft track to the ex- 
treme positions of the submarine (6 or c, as the case 
may be). In equation (22), a (i.e., h or c) may be 
greater than wT and the function h may be greater 
than unity. Nevertheless, as long as d is less than 
wT, subtracting these h’s will give the correct prob- 
ability. Working from the g{E'/a) curve, h[{E'/wT), 
(a/wT)] was calculated for various a/wT and plotted 
against E’ jwT on Plate I. Obtaining p is just a matter 
of reading, adding or subtracting values from the 
curves. 

If d is greater than wT, a given submarine may be 
exposed to more than one aircraft pass. In the sim- 
plest such case the submarine path extends exactly 
to the point where it intersects the aircraft track and 
d = 2wT. For the rigorous treatment of this case, a 
submarine which is at distance x < wT from 0 on 
one pass must have been exposed at the position 
X + wT on the previous pass. Equation (12) then 
becomes, 

V2wT = ~ (x ( 26 ) 

This is plotted as curve u on Plate I. 

There are no general formulas giving rigorous so- 
lutions when d > wT. Each situation would have to 
be treated separately and usually with more difficult 
integration than in equation (26). 

The following approximate procedure has been 
found sufficiently accurate. A submarine’s position 
in one wT segment is assumed to be independent of 
its position in the adjacent one at the time of the 
previous aircraft pass. The submarine path BC can 
then be broken into sections Si of length wT or less; 
the probability of contact can be found for each sec- 
tion using equations (22) or (23). Finally, these 
probabilities, say pi, p 2 • • * Vn, can be combined as 
independent events, to give p^c- This may be ex- 
pressed by : 

p = 1 - (1 - pi) (1 - Pi) (1 - ps) • • • (1 - P,)(27) 

Plate II gives a rapid solution of equation (27) for 
two p’s; for more sections Plate II may be used re- 
petitively. 


The series oi wT sections is placed with respect to 
BC in such a way that one of the wT sections has as 
high a chance of contact as possible. Then the con- 
tribution of the other sections is minimized and so 
is the effect of the independence assumption. For 
Figure 11 the principal section. Si, should be taken 
from C to C + wT. In the special case of Figures 9 
or 10, Si would be from 0 to wT. In Figure 12A, if h 
and c are greater than ll2wT, Si should extend from 
— ll2wT to l/2wT. If c < ll2wT, Si should ex- 
tend from — c to wT — c. If as in Figure 12B the 
aircraft turns at 0, the larger part of Si should be on 
the side of 0 having the best chance of sighting, that 
is, the side associated with the larger E'] usually Si 
was taken from 0.2wT on the weak side to O.SwT 
on the strong side. 

To find the order of magnitude of the error in- 
volved, consider the case treated rigorously in equa- 
tion (26) in which the submarine may be sighted be- 
tween X = 0 and x = 2wT. Curves u and v of Plate 
I give, respectively, the rigorous and the approximate 
solutions. These differ by only about one per cent, 
which is a smaller error than those to be discussed in 
the following paragraphs. The possibility of fluctua- 
tions in the values of w and T introduces doubts with 
regard to the rigorous method and makes the ap- 
proximate method even more excusable. 

Now that we can handle any interval of submarine 
track, let us begin to consider the effectiveness of an 
actual screening plan against a submarine on a 
specific collision course. Unless detected, the sub- 
marine proceeds from well beyond detection range to 
an end point E at which it either submerges or fires 
at the convoy. Also relative to the convoy, the air- 
craft will repeat every T hours a closed path com- 
posed of straight legs. None of these legs goes to 
infinity. Nevertheless, the only practical way of treat- 
ing a screening plan is to divide the aircraft path into 
sections each of which can be treated as an approxi- 
mation to one of the cases already discussed. In each 
of these cases the aircraft path was an infinite 
straight line coming up to the submarine track and 
the same or another line going off to infinity. 

Consider the illustrative aircraft path FGHIJF 
and the submarine path in to E as shown in Figure 
15. In this case four hypothetical aircraft paths must 
be considered; three of them are made by extending 
the legs FG, GH, and JF to infinity in both directions; 
on the fourth path the aircraft approaches from a 
great distance on a straight path through H to I 
where it turns and goes out to infinity on a straight 


154 


AERIAL ESCORT 


path through J. Each infinite path is to be treated 
by one of the methods described earlier to give the 
probability of contact if the plane were searching 
only during the corresponding segment, FG, GH, 
HIJ, or JF, of the screening plan. (These individual 


\ 

\ 



/ 


Figure 15. Example of path and track approximations. 

probabilities will later be combined to give the ef- 
fectiveness of the entire plan.) However, if with each 
infinite path the entire submarine track in to E were 


considered, the individual probabilities found would 
be too high, as a result of the finite chance of con- 
tact while the aircraft is on the imaginary extensions. 
This error is to be eliminated roughly by limiting in 
each case the section of the submarine track in which 
a contact can be made. Thus an infinite aircraft 
track and a limited submarine track shall be sub- 
stituted for a limited segment of aircraft track and 
an infinite submarine track (see Figure 16). This 
substitution is required because only infinite aircraft 
tracks can be handled at all simply. 

The following scheme is employed in limiting the 
submarine track. The submarine is never allowed be- 
yond the limits set up in the original problem, i.e., 
to the left of E in Figure 15. 

Let h' and c' be the distances from 0 (the inter- 
section of the aircraft and submarine paths or their 
extensions) to the limits of the segment of aircraft 
track. Whenever possible h and c, the distances from 
0 to the extremes of the submarine path, are taken 
to equal h' and c', respectively, and are chosen so 
that the angles between b and b' and between c and 
c' are each acute. For example, in Figure 15 the sub- 
marine intervals F'G', EG", and EF" are used for 
legs FG, GH, and JF, respectively. The principal ex- 
ception occurs when, as with HIJ, the side of the 
submarine track makes an obtuse angle with both 
legs of the aircraft track; then that side of the sub- 



Figure 16. The geometrical principles of path and track approximation. 


INSTRUCTIONS FOR OBTAINING A POLAR DIAGRAM 


155 


marine track is not limited, while the other side must 
be limited in length to a weighted average of the 
lengths of the aircraft legs. In Figure 15 this ques- 
tionable averaging is avoided and the entire sub- 
marine path (to the right of E) can be used, since 
IE is less than either IH or IJ. 

The last paragraph may be justified as follows: In 
the real situation with the aircraft on a limited seg- 
ment of track, most of the chance of contact accu- 
mulates while the submarine is on the part of its 
track to which it is limited in the approximation, be- 
cause in most cases this is the part of the infinite sub- 
marine track closest to the segment of aircraft track. 
This large portion of the desired probability is cor- 
rectly obtained by the approximate method, for it 
is the part contributed by the portion of the infinite 
aircraft track which lies within the real segment. The 
remaining so-called ‘‘end effects’’ are in the real case 
the chance of the aircraft on its segment contacting 
a submarine outside BC (in Figure 16A) and in the 
approximation the chance of the aircraft on the in- 
finite extensions of its segment contacting a sub- 
marine in BC (Figure 16B). The symmetry about 0 
between the real and the approximate cases makes 
the end effects about equal, since all the submarine 
to aircraft distances are duplicated. A systematic 
error (a pessimistic one) enters only when the air- 
craft segment is much shorter than wT. The error 
due to terminating the submarine track at a point 
E within BC enters only into the end effects and is 
usually small. In any case, these errors are small 
compared with those which could arise from the in- 
accuracy of the inverse cube law. 

The chances of contact on the individual segments 
of aircraft track are combined as independent prob- 
abilities to yield the desired probability of detection 
by the entire plan. Plate II may be used in making 
this computation. The error resulting from the inde- 
pendence assumption is seldom over four per cent. 
Usually one of the individual probabilities predomi- 
nates and so the method of including the others is 
unimportant. 

In initially choosing the submarine collision courses 
(i.e., the 6 for which p{6) is computed), it is ad- 
visable to take one through each corner of the air- 
craft track and then avoid others close to these 
corners. Those through the corners can be handled 
well by the bent track formulas given above. On the 
other hand, with neighboring tracks, the two legs 
(such as EG and GH at G in Figure 15) have to be 
extended to infinity. The farther the intersection is 


from the submarine track, the less important will be 
the end effects due to these extensions. Also the in- 
dependent combination of the probabilities on these 
two legs is a dubious procedure since both legs have 
their best chance of contact against the same part of 
the submarine track at the same time. The farther 
G is from the submarine track the better. 


9 6 practical instructions for ob- 
taining A POLAR DIAGRAM^ 

The foregoing will now be summarized in a set 
of practical rules for obtaining a screening polar dia- 
gram.® Illustrating these rules, Plate III contains the 
calculations leading to the polar diagram (Figure 18) 
for the air escort plan of Figure 17. 

1. Starting with the aircraft track drawn relative 
to the convoy as shown in Figure 17, choose the sub- 
marine paths to be studied in evaluating the plan. 

a) . Each submarine path starts well outside 

sighting range and comes in on a collision 
course until it reaches the danger zone (sub- 
merged or firing area). 

b) . Each path is associated with the angle 6 

which it makes with the convoy’s heading. 

c) . There should be a sufficient number of paths 

so that no important variations in the polar 
diagram will be missed. 

d) . The submarine paths through each corner of 

the aircraft path should be included. 

e) . Other paths close to corners should be 

avoided. 

f ) . For escort plans symmetrical about the con- 

voy’s line of advance, only 6’s up to 180 de- 
grees need be considered. 

2. Make out a table with column headings 1 to 
20, as in Plate III. 

3. Record the d’s and corresponding values of w, 
the submarine’s speed relative to the convoy. 

4. Using T, the time (hours) of repetition of the 
escort plan, calculate and record the values of wT. 

a). In the absence of a true space diagram, it is 
sufficiently accurate to calculate T by di- 
viding the length of the relative aircraft 
track by its true speed. 

5. For each submarine path divide the aircraft 

g In the method given here, the probabilities p(0) are computed 
for paths ending not at the convoy reference point 0 , as in 
the general discussion of Section 9.4, but at the boundary of 
the submergence region (here, the danger zone). 


156 


AERIAL ESCORT 



Figure 17. A possible aircraft escort plan. 



INSTRUCTIONS FOR OBTAINING A POLAR DIAGRAM 


157 



Figure 18. Screening polar diagram. 


158 


AERIAL ESCORT 


track into segments each of which is either a single 
straight leg or two legs joined by a turn on the sub- 
marine’s path. 

a) . A row of the table should be devoted to each 

of these segments. 

b) . Usually there are segments located so far 

from the submarine path that their contri- 
bution is negligible and they can be elimi- 
nated by inspection. 

6. Columns 4 to 8 deal with pertinent data meas- 
ured from the relative diagram and are best filled 
out row by row (see below). 

7. Record in column 7, for each straight segment, 
the angle a between the segment and the submarine 
path or their extensions. 

8. Record in column 8, for each bent segment, the 
angles 71 and 72 between the submarine path and 
each leg of the segment. 

a). Both 7i and 72 are measured from the same 
side of the submarine track from the inter- 
section. 

9. For each segment of aircraft track the sub- 
marine path is to be limited, in order to compensate 
for substituting an infinite aircraft track for the 
actual segment. 

a) . The distances h and c (as used in columns 4 

and 5) are measured from 0, the intersection 
of the aircraft and submarine paths or their 
extensions, to the extremes, B and C, of the 
limited submarine path. 

b) . With the exception noted below, h and c are 

equal to the distances from 0 to the extremes 
of the aircraft segment; B and C are each 
found by rotating the segment about 0 
through an acute angle to coincide with the 
submarine path. Exception. When the air- 
craft turns so that the 7’s are either both 
acute or both obtuse, the part of the sub- 
marine path making obtuse angles with both 
legs is not limited. The other part is limited 
to an average of the two legs (weighted by 
the cosecants of the 7’s). 

10. In columns 4 and 5 record h/wT and cjwT. 

a). If BC is entirely on one side of 0, the 

smaller of b/wT and c/wT is recorded in 
column 5 and given a negative sign, by 
convention. 

11. Whenever the aircraft segment and submarine 
path are approximately parallel, B and C are found 
as in 9 above, where in this case the rotation is about 
a point 0 at a great distance. 


a) . The length BC of the limited submarine 

path is called d and divided by wT for 
listing in column 6. 

b) . Also listed in column 6 is x, the average 

distance between BC and the aircraft seg- 
ment. 

12. Record in columns 9 and 10 the required trigo- 
nometric functions. 

13. Record the following in columns 11 and 19. 

a) . For straight segments, 

E'/E = CSC a 

[in accordance with equation (15)]. 

b) . For bent segments, 



^ = 3^ Vcsc2 7 i/ 2 + csc2 72/2 

and 



M Vsec2 7 i /2 + sec2 72/2, 

corresponding to the two parts of the sub- 
marine path separated by the intersection 
point 0. The former applies to the part of 
the path from which the 7’s are measured. 
[Equations (16) and (20)]. 

c) . Only one of the equations in b) above is 

used when the limited submarine path falls 
entirely on one side of 0. 

14. Columns 13 to 20 involve the value of E, the 
effective visibility, and must be recalculated for each 
such value. 

15. For flights parallel to the submarine path, 

a) . Record in column 13 the values of E/x, 

using the entry that has been made in 
column 6. 

b) . Obtain Pparaiiei from Figure 14 by using E/x. 

c) . Whenever d/wT ^ 1, obtain psc (column 

19) by multiplying Pparaiiei by d/wT. (psc 
is the probability associated with this seg- 
ment of the aircraft track against the sub- 
marine path.) 

d) . Whenever d/wT = n f, where n is an 

integer and 0 ^ / < 1, multiply Pparaiiei 
by / to obtain pf. Using equation (27) or 
Plate II, obtain psc by combining Pparaiiei 
with itself n times and with pf all as inde- 
pendent probabilities. 

16. Record the values of E'jwT in columns 14 and 
15. These are obtained by multiplying EJwT by the 
entries, E'jE, in columns 11 and 12. 


Plate III. Calculation of the Screening Polar Diagram for Plan K. 


INSTRUCTIONS FOR OBTAINING A POLAR DIAGRAM 


159 


Assuming £' = 6 miles 

o E 

^ ft. 

0.245 

0.25 

0.27 

0.41 

0.32 

0.34 

1 

0.37 

0.44 

0.49 

990 

0.54 

610 

2 ^ 
a 

0.124 

0.005 

double 

0.144 

0.107 

0.015 

O 

00 O 

doo 

0.342 

0.107 

0.202 

0.125 

0.024 

0.006 

iC ^ 

05 CO l>- 

^ ^ o o 
dddd 

891' 0 

mo 

0.270 

0.203 

0.018 

0.298 

0.272 

0.375 

0.44 

0.035 

0.000 

0.465 

0.101 

0.040 

0.090 

0.112 

18 

Additional 
contributions 
from other 

wT’a 







O 

O 

010 0 


o 


’H ‘>y 

Ll 91 

V 

a 

0.062 

-0.150 

o 

o 

o o 

OSO 0 

060 0 

o o 

d d 

90 0 

90 0 

kZOO 

080 0 

0.062 

0.080 

0.135 

0.092 

-0.180 

0.145 

0.127 

0.200 

0.22 

0.05 

-0.995 

-0.335 

-0.213 

-0.45 

a g 

o 

0.062 

0.155 

0.72 

0.057 

ssoo 

060 0 

0.300 

0.057 

0.142 

0.065 

0.115 

0.06 

0.175 

0.106 

0.135 

0.111 

0.198 

0.145 

0.145 

0.175 

0.22 

0.415 

1.096 

0.375 

0.303 

0.562 





00 

o 

d 



0.124 




oro 



0.125 

0.306 

0.145 

0.115 

iiro 

0810 

0.704 

0.114 

0.302 

0.131 

• 

0.241 

0.148 

0.382 

0.212 

0.282 

0.223 

0.396 

0.31 

0.308 

0.402 

0.51 

0.200 

00 05 
O5C0 

O C^* O 

ZZl 
89 0 

CO 

- aq 


0.273 

0.75 


0.414 

0.545 

0.75 

0.316 




0.667 



Xfl 

13 

O 

00 

CO 

(N 

II 

10 

00 <M 

1^ 

11 

Sh 

o 

C 

II 

x> 

g 

02 

O 

C 

O 

II 

» 

05 

o 

c 

o 

(M 

T— H 

II 

Si 

II 

<M ^ 

+ 

C<J 

> 

HN 




0.72 



0.76 




0.76 


II 

V) 

+ 

cT 

1.19 

2.92 

1.37 

1.087 

1.66 

1.027 

6.15 

1.001 

2.36 

1.021 

1.74 

1.07 

2.34 

1.304 

1.347 

1.062 

1.888 

1.021 

1.017 

1.05 

1.304 

5.24 

05 40 05 
t>. 00 

(M 4C '-1 

1.49 

2.92 

o 

1 




1.003 1.027 



1.030 1.118 




1.033 1.111 


^|<M 

8 

O 

o 

2 




11.47 4.45 



4.13 2.24 

1 



3.86 2.28 


8 

71 72 

(degrees) 




10 26 

1 



28 53 




30 52 


7 

a 

(de- 

grees) 

o 

»0 (M 

t>« 

ec !>• 

00 

iC 00 

05 

CO O 


00 o 

^ CO 

00 

O ^ 

lO ^ 

O C<l 
^ CO 

<M O 

6 

d/wT X 
miles 
(For parallel 
paths) 


0.53 22 

0.54 8 


0.6 14.5 

0.13 11 

0.35 8 

0.7 19 




0.54 9 



5 

c/wT 

0.175 

-0.65 

0.221 

0.12 

0.335 

0.10 

00 

0.095 

0.065 

0.11 

^ 05 

d d 

00 

0.12 

0.42 

0.14 

-0.54 

0.45 

0.24 

0.61 

0.45 

-3.0 

00 

-1.8 

-0.65 

-0.28 

-0.63 

4 

b/wT 

0.473 

1.14 

0.37 

0.565 

0.236 

0.643 

0.572 

0.8 

0.60 

0.26 

0.76 

0.16 

0.53 

1.0 

0.4 

1.4 

1.14 

1. 

0.58 

0.33 

0.43 

3.5 

0.84 

2.55 

1.35 

1.05 

1.43 

3 

wT 

(miles) 

57.1 

56.6 

55.2 

52.5 

46.9 

43.3 

36.8 

28.6 

00 

05 

15.7 

14.5 

14.3 

2 

w 

(knots) 

1 

23.8 

23.2 

22.1 

19.7 

18.2 

15.5 

12.0 

8.3 

6.6 

6.1 

6.0 

1 

e 

(de- 

grees) 

o 

O 

20 

30 

45 

55 

70 

06 

120 

150 


180 



160 


AERIAL ESCORT 


17. When the sum of the values in columns 4 and 
5 is less than or equal to 1, 

a) . These values together with those in columns 

14 and 15 may be used together with Plate 
I to yield and he (columns 16 and 17). 

b) . Giving column 17 the same sign as column 

5, add algebraically and he to obtain psc 
(column 19). 

18. When the sum of the values in columns 4 and 
5 is greater than 1, 

a) . Choose the section S (length wT) of sub- 

marine path BC, which has the greatest 
chance of contact. Within the limits im- 
posed by columns 4 and 5, the center of S 
should be located as follows : 

(1) For straight aircraft segments the cen- 
ter should be as close as possible to the 
intersection point 0. 

(2) For bent aircraft segments, the center 
should be as close as possible to a point 

O.ZwT from 0 on the side having the 
greater E'/E. 

b) . Using the a/wT’s corresponding to the ex- 

tremes of S and using the E'lwT’s from 
columns 14 and 15, find the corresponding 
h’s from Plate I, and enter them in columns 
16 and 17. 

c) . For each of the remaining sections (length 

wT or less) of BC, find the chances of con- 
tact (by a subtraction of two points having 
the same abscissa on Plate I) and enter 
these separately in column 18. 

d) . Obtain psc (column 19) by combining the 

algebraic sum of columns 16 and 17 with 
these smaller probabilities (column 18) by 
the independence procedure [equation (27) 
or Plate II]. 

19. Column 19 now has the contribution of each 
aircraft segment against each submarine path. For 
each submarine path combine these Pbc’s as in- 
dependent probabilities [using equation (27) or Plate 
II] to give pid), the chance of contacting the sub- 
marine on collision course 6 when the whole aircraft 
plan is considered. 

20. Plot p(d) against 6 to give the desired polar 
diagram (Figure 18). 

9 7 the design of a plan 

As in all questions of this nature, there are two 
possible viewpoints. Either we may lay down the 


tactical results to be achieved (scouting and screen- 
ing) and then find a plan which achieves them with 
the least expenditure of effort (number of aircraft 
and flying time), or else we may fix the total amount 
of effort available and seek the plan which maximizes 
the tactical results. Now the solution of the second 
problem furnishes that of the first, since it auto- 
matically informs us of what can be achieved with 
one aircraft, with two aircraft, etc.; and it remains 
only to pick the first plan in this sequence which 
gives the required result. It shall accordingly be from 
this viewpoint that the problem is approached here. 

Let the number n of aircraft be given together 
with their capabilities of speed v, endurance, and de- 
tection. The convoy’s speed Ve (in the sense of mean 
course made good) is also supposed to be known. 
Before there can be any question of the “best” plan, 
a decision must be made as to how much relative 
importance must be attached to scouting as compared 
with screening. Now this is a purely tactical question. 
It must be settled on the basis of presumed enemy 
tactics and submarine capacities, as well as of our 
own defensive capabilities and our vulnerability. For 
example, against a submarine capable of high sub- 
merged speed and endurance, screening could be 
expected to be less effective than scouting, whereas 
the reverse might be true if the submarine were of 
the older type without these capabilities. When this 
decision has been reached and the relevant speeds 
and distances have been estimated, five conditions 
must be satisfied. 

1. The entrance to the submerged approach region 
must be adequately guarded. 

2. Flying must not be unduly concentrated about 
the convoy. 

3. The circuits must be closed in convoy space, 
i.e., the aircraft must automatically meet the con- 
voy. 

4. The time between successive meetings of the 
convoy must not be excessive (never more than two 
hours; one hour is much better than two). 

5. The plan must be navigable with reasonable 
ease. This means that one involving many turns must 
be avoided. 

Now obviously it is not feasible to deduce an exact 
plan from these data and the requirement of maxi- 
mizing scouting and screening. It is necessary first 
to invent plans and afterwards to test, modify, and 
select until a satisfactory one is obtained, and to ex- 
haust all visible possibilities of improvement. Such a 
procedure is an art quite as much as a science. It 


THE DESIGN OF A PLAN 


161 


BASE COURSE 



LEFT-HAND A/C RIGHT-HAND A/C 


BASE 

LEFT-HAND AIRCRAFT 

.0:00 


COURSE 

RIGHT-HAND AIRCRAFT 



Plate IV. Actual tracks of aircraft in operational flight. Inset shows search plan being used. 


162 


AERIAL ESCORT 


is advisable to make the designs first relative to the 
convoy rather than in space fixed with respect to the 
ocean, since the former picture is simpler and more 
direct in its illustration of relevant features; but 
eventually it will be transferred to geographic space, 
where the last refinements can be made. 

When a satisfactory plan has been designed for one 
speed ratio r = v/vc, it is usually desirable to extend 
it without changing its fundamental character, so 
that it will be applicable to broader ranges of r. This 
should be done without changing the angles of the 
courses in the original plan, if possible, but rather by 
varying certain leg lengths. It is simply the kine- 
matic problem of bringing all planes back to the con- 
voy periodically every T hours, given the new values 
of r. It is solved by simple trigonometry as follows: 
Draw a full period of the plan in geographic space, 
i.e., the figure that repeats itself every T hours. Con- 
sider the part of the plan flown by one aircraft. Let 
it be desired to vary two particular leg lengths, x and 
y, maintaining the others constant (values given). Let 
z be the distance the convoy moves in the time T. 
Then the three variables x, y, z are related by three 
equations of the first degree: first, the requirement 
that the length of the flown path {vT) divided by z 
{vcT) be equal to v/vc = r; second, that the algebraic 
sum of the projections of the legs on the convoy path 
shall be z] and third, that the algebraic sum of the 
projections on a normal to the convoy path be zero. 
Solving these equations, x, y, z are obtained as func- 
tions of r and, of course, T. Tables of the results are 
furnished with the plan. Of course, it is necessary to 
re-examine the plan (calculating scouting and screen- 
ing diagrams anew, at least at critical places) for 
the extreme values of r used, in order to be sure that 
the necessary protection is maintained. 

Since the effectiveness of a plan depends on the 
conditions of detection (visual or radar), reasonable 
and conservative estimates of visibility should be 
made in making all these calculations. It may be 
necessary to repeat such calculations for other visi- 
bilities. However, it is unlikely that a plan which is 
better than another at one visibility shall be worse 
than it at a different visibility, unless the change of 
visibility entails an essential change in the tactical 
situation. 

It may be remarked that the diagram in Figure 
18 exhibits an imperfection in the plan of Figure 17, 
namely, insufficient protection in the important for- 
ward regions. Better plans are actually used. 

An important point of a practical nature must be 


kept in mind in designing and evaluating any air 
escort plan. It is that aircraft are not apt to fly a 
given plan to a very great degree of accuracy. Just 
how far they may deviate from the plan will depend 
on wind, visibility, radar and other navigational 
equipment, as well as on the temperament and ex- 
perience of the pilot. It will depend also on his find- 
ing it necessary to investigate presumed contacts, 
which very often turn out to be false contacts. The 
chart shown on Plate IV gives the actual flights 
(drawn relative to the convoy) of two aircraft which 
were flying the simple straight-line plan shown in 
the insert (a plan which is now obsolete). The air- 
craft were actually tracked by radar from the carrier 
during an operation in World War II and their posi- 
tions marked at the successive epochs indicated on 
the chart. This provides an object lesson on the 
difference between theory and practice. Incidentally, 
it explains why it is so often realistic to apply the 
formula of random flights of Chapter 2, equation 
(40). It has even been felt that, after all, the chief 
value of having a systematic plan is to get the (per- 
force, random) flights out where they would be ef- 
fective, rather than to leave the pilot to follow his 
own devices, which has generally led to flights 
bunched up in the wrong places, usually too close to 
the convoy. 

98 FINAL SWEEPS 

As long as the convoy is in submarine waters, 
maximum protection is obtained by flying the escort 
plan without letup. But it is often necessary to dis- 
continue the flights for a protracted period of time, 
for example, during the hours of darkness. The 
danger incident to this can be minimized by flying 
final sweeps at a greater distance than the normal 
flights, immediately after the latter are discontinued. 
These final sweeps are essentially of a scouting char- 
acter; they aim at detecting submarines which might 
constitute a menace to the convoy at a later period, 
after aerial escort has been discontinued. 

The first thing to realize is that it is not normally 
possible to detect by flights made at the time (t = 0) 
of discontinuance of aerial escort, all the submarines 
which could possibly close the convoy during the 
next H hours of escortless travel. For it would be 
necessary to sweep the area of the ocean in which 
such submarines must be; this, in the case of w > Vc, 
is the shaded circular region of Figure 19, obtained 


FINAL SWEEPS 


163 


by multiplying all lengths of the circular diagram of 
relative speeds (vector w) by H. This means that an 
area of would have to be swept. For example, 

if w = 12 knots and H = 8 hours, the area is 29,000 
square miles. With an aircraft of 130 knots and sweep 
width 10 miles, a time of 29,000/1,300 = 22.3 hours 



Figure 19. Circle of possible future contacts. 


would be required. Even with four planes, 5.6 hours 
would be needed; and more, if the difficult track 
problem of covering the area is taken into account, 
with the resulting loss of efficiency. If less than this 
coverage is available, what should be covered and 
what neglected? This is essentially the question 
treated in Chapter 3 (specifically, the corollary at 
the end of Section 3.3). 

Taking a system of rectangular coordinates with 
origin fixed at the convoy reference point 0 (and 
hence moving with the convoy), and axis of ab- 
scissas in the direction of convoy motion, we must 
evaluate the function p{x,y), where p(x,y) dxdy is the 
probability of there being a “dangerous’’ surfaced 
submarine in a dxdy region at the point {x,y) (at the 
initial epoch ^ = 0), where by a “dangerous” sub- 
marine is meant one which will surely come in con- 
tact with the convoy (i.e., enter the detection circle 
of radius R, center 0) during the subsequent H 
hours. Next, we must evaluate the amount of cover- 
age or searching effort available. Thus if we have 
the use of n aircraft for h hours each and if their 
speed is v and effective search width is IF, $ = nvhW. 
According to the reasoning of Section 3.3, we are 


warranted only in searching the region of ocean A 
characterized by the two following properties : 

1. Ajj is the locus of points (x,y) for which p{x,y) 
^ h; this determines the shape of A 5. 

2. The value of b and therewith the choice of the 
region A 5 are fixed by the requirement of equation 
(22) of Section 3.3. According to Chapter 3, A^, 
should then be searched in the manner specified by 
equation (23) of that chapter. A geometric construc- 
tion is given therein. But in practice, such refinement 
is not apt to be possible, and the best procedure 
would be to use any practicable search, of a more or 
less uniform sort, in the estimated region A 5. Of 
course the whole of the detection circle itself must be 
included in the search to detect submarines already 
in contact with the convoy, particularly trailing be- 
hind and tracking on the flanks. 

The estimation of p{x,y) is difficult, not because 
of the mathematics involved (a crude mathematical 
formulation is quite sufficient) but because it de- 
pends on the correct appraisal of the tactical situa- 
tion regarding the submarine. There are two extreme 
cases which may be considered as examples. 

Case 1. The submarines are uniformly distributed 
over the part of the ocean of interest and are cruis- 
ing at an estimated speed of u knots in any direction 
(i.e., in uniformly distributed random directions; see 
Section 1.4). This is the case in which the submarines 
are picked up purely by chance and not as the result 
of any systematic patrolling on their part. The prob- 
lem of finding p(x,y) in this case has been solved in 
Chapter 1 [see Section 1.5, problem 3, together with 
Figures 10, 11, 12, and 13; the 4>(r,/3) of this reference 
is not to be confused with the “amount of searching 
effort” 4> used above]. In this reference, the prob- 
ability P{r,A) is found [(r,/3) are the polar coordinates 
of {x,y)\] unlike p{x,y), this is not a probability or 
expected value density but a probability, namely, the 
probability that the target given to be at (r,|3) [or 
{x,y)] when t = 0 shall subsequently enter the detec- 
tion circle. But if the mean density of submarines is 
N, there will be Ndxdy submarines in all in the dxdy 
region at {x,y), of which the fraction P{r,A) will 
enter the circle; hence the relevant expected number 
of submarines in NP{r,A) dxdy = NP{r,A) rdrd^. Now 
as in the proof of corollary at the end of Section 3.3, 
the constant of proportionality N is immaterial in 
the final results. Hence equations (22) and (23) as 
well as the geometrical construction given in that 
chapter apply with the former p(x,y) replaced by 
P{r,^). Finally, in the problem of Chapter 1 the time 


164 


AERIAL ESCORT 


before contact was not limited, whereas we are re- 
stricting it to H hours. But with the searching effort 
normally available, only regions within an H hour 
submarine run of the detection circle can be searched, 
so that the time restriction applies automatically. 

Let us consider as example the case Vc/u = k = 
e.g., submarine cruising at 15 knots, convoy at 

10 knots. The level curves of constant probability 
P(r,i3) (i.e., the boundaries of for different values 
of h) are given in Figure 11 of Chapter 1. Let the 
detection range of the submarine on the convoy be 
R = 20 miles. This gives the scale in Figure 11, 
Chapter 1, of 20 miles to the inch. With two 130- 
knot aircraft available for two hours and assuming 
the search width IF = 10 miles, we have ^ = 
2 X 2 X 130 X 10 = 5,200 square miles of searching 
effort available. To determine the probability con- 
tour of Figure 11, within which the search must take 
place, we have to invoke (22) of Chapter 3, which 
becomes 

log P(r,j8)rdrd/3 — A^logh = ^ = 5,200. 

Ab 

The values of P{r,^) ior k = % would have to be 
drawn from Section 1.5, Chapter 1, and the inte- 
gration and calculation of the area performed for 
different values of h, the arc finally chosen. Then a 
plan of search would be devised as much as possible 
in accordance with (23) of Chapter 3. 

The following rough graphical version of this 
process illustrates the principles of the method in a 
form which can be carried out without unreasonable 
trouble in practice and which provides about all the 
accuracy which our rather dubious tactical and nu- 
merical assumptions would appear to warrant. 

With a planimeter (or an approximating ellipse) 
calculate the areas of the various Ab regions of Figure 

11 (Chapter 1), remembering that h = 0.1 for the 
10 per cent region, h = 0.15 for the 15 per cent, 
• • *,5 = 1 for the detection circle. We find approxi- 
mately for the A areas, and the AA rings between 
successive regions: 


Ai 

= 1,260, 

AAi 

= Ai - 

Ao.4 

= 

150, 

Ao.4 

= 1,410, 

AAo.4 

= Ao.4 “ 

Ao.25 

= 

1,710, 

Ao.25 

= 3,120, 

AAo.25 

= Ao.25 “ 

Ao.15 

= 

3,030, 

Ao.15 

= 7,050, 

AAo.15 

= Ao.15 ~ 

Ao.io 

= 

8,950, 

Ao.l 

= 16,000. 






To find 

the graph 

of the 

function 






Ab 


against h out to 5 = 0.1, we may multiply each ring 
area by the arithmetic mean of the (natural) log- 
arithms of the probabilities of the two bounding level 
curves (except initially, the area of the detection 
circle is multiplied by log 1=0), and then add the 
results out to the ring bounded by the value of h in 
question. The curve for f{h) is then drawn through 
the resulting five points. Next, the function 

F(b) = 5,200 + Afe log h 

is graphed, taking the five values of A 5 given above 
and multiplying by log b, h being the value corre- 
sponding to the boundary; a smooth curve is then 
passed through the resulting points. These curves 
are found to intersect at about the point h = 0.13, 
i.e., /(0.13) = F(0.13); but this is equation (22) of 
Chapter 3. 


Table 1. 




Avg log 

A 

Avg log 



b 

log h 

b 

(Ai = 1,260) bXAAb 

f(b) 

m 

1.0 

0 

0 


0 

0 

5,200 

0.4 

-0.91 

-0.45 

150 

68 

68 

3,910 

0.25 

-1.49 

-1.20 

1,710 

- 2,060 

- 2,128 

550 

0.15 

-1.89 

-1.69 

3,930 

- 6,650 

- 8,778 - 

- 8,100 

0.10 

-2.20 

-2.04 

8,950 

-18,300 

-27,078 - 

-30,000 


Now since one cannot hope to search in accordance 
with equation (23), Chapter 3, the best practical 
recommendation is to cover a region extending a 
little beyond the 15 per cent curve of Figure 11 
(Chapter 3), for example, by a sector search. The 
probability of detecting the target by a random 
search of the 15 per cent curve, is, by (40) of Chapter 2, 

p = 1 - = 1 - “ 

= 1 _ ^-5.200/7.050 ^ Q 52. 

Thus, there is a little over half a chance that a 
given submarine in the 15 per cent region will be 
detected. Therefore, the search is decidedly useful, 
but by no means as good as having regular escort 
flown through the night. 

Case 2. The submarines are mounting guard along 
the route of the convoy, as they estimate it. If there 
are several submarines acting as a coordinated group, 
they may form a line patrol across the convoy path 
of a type illustrated by the following example : Sub- 
marines are placed in a line across and at right angles 
with the convoy path patrolling stations 20 miles 
apart; each submarine proceeds at a low speed 
('w = 10 knots) and is never more than half an hour’s 
run from its station. [This was a German plan de- 



FINAL SWEEPS 


165 


scribed in the British Monthly Anti-Submarine Re- 
port (secret), November 1942.] Submarines acting 
alone may patrol at greater distances about their 
stations, but in all cases the distance is limited by 
the objective of intercepting convoys moving along 
the line. 

This case is easier to treat than the first, since the 
only dangerous submarines here lie in a band whose 
middle line is the convoy’s future course and whose 
total width is 2{R -f d), where d is the presumed 
distance of lateral patrol of the submarines, while R 
is (as before) the submarines’ detection range on the 
convoy during the time of discontinuance of the 
normal aerial escort (e.g., at night). For a submarine 
at lateral range more than R d on either side of 
the convoy’s course would be unlikely to constitute 


a threat; p{x,y) = 0 outside the band, p(x,y) = 
constant > 0 within the forward length of the band 
of Hvc miles. Thus, this whole piece of band must 
be swept. This is a feasible operation. With R = 20, 
d = 10, V = S, H = S, the rectangular band’s area 
is 3,840 square miles; one plane of 130 knots and 
10-mile search width would require 3,840/1,300 = 
2.95 hours. With two or three planes, not only this 
area but a somewhat wider one, including the day- 
time detection circle, could easily be swept in a very 
reasonable time, as the last hour or so of daylight. 

But in setting up this search, it is essential to know 
in advance what deceptive steering is intended dur- 
ing the night. Convoys usually make a deceptive 
change of course shortly after dark; the band may 
have to be a bent one under such circumstances. 



DEGRADED UNCLASSIFm 
oitm ssc Amy by t ag 1 On i 





GLOSSARY 


A/C. Aircraft. 

Acoustic Torpedo. Homing torpedo guided to its target by 
means of echo ranging or listening. 

Anomalous Propagation. In sonar, pronounced and rapid 
variations in echo strength caused by large and rapid local 
fluctuations in propagation conditions. 

ASB Radar. A 60-cm Navy radar for surface search by carrier- 
based aircraft. 

Asdic. British echo-ranging equipment ; letters are derived from 
“Anti-Submarine Development Investigation Committee.” 

ASG Radar. AN/APS-2, a 9-cm ASV and search radar. 

ASV Radar. A radar system for detecting and homing on a 
surface vessel from the air. 

Asymmetry Factor. Ratio of target length to width. 

Barrier Line. Mathematical reference line across and per- 
pendicular to a channel. 

Bearing. Angular position of target with respect to own ship 
(relative bearing) or to true north (true bearing) . 

Blip. In radar, echo trace on indicator screen. 

Brightness Contrast. The difference between target and im- 
mediate background brightness expressed in units of effec- 
tive background brightness. 

Browning Shot. A shot aimed at a general area containing 
targets, on the chance of a random hit. 

B-Scope. a scope which presents a rectangular plot of range 
versus bearing. Spot brightness indicates echo intensity. 

Cavitation. The formation of vapor or gas cavities in water, 
caused by sharp reduction in local pressure. 

Channel. Strip of ocean through which it is known targets 
must pass. 

Collision Course. Course steered by an attacking craft to 
intercept the attacked craft’s course so that a coincidence 
will occur between the two crafts. 

Contact. An instance of detection of an enemy unit. 

Contact Problem. Study of the capabilities of the detection 
agency. 

Convoy. A group of merchant ships sailing together, usually 
defended by naval craft escorts. 

Curly Torpedo. A torpedo which, after closing the convoy, 
steers a sinuous course to increase its chance of a random hit. 

CVE. Navy designation for Aircraft Escort Carrier, a small 
aircraft carrier. 

Definite Range Law of Detection. The assumption that 
detection is sure and immediate within a given critical range 
and impossible beyond that range. 


Density. Number of objects per unit area. 

DE. Destroyer escort, a large, high-speed antisubmarine ship. 

DF. Radio Direction Finding; use of directional receivers to 
estimate enemy position by intercepting and obtaining the 
bearings of enemy transmissions. 

Echo Ranging. Method of locating underwater objects by 
sending sound pulses into the water and receiving an echo 
reflected from the target. Target range is derived by measur- 
ing total transit time of the sound pulse. 

Effective Radar Cross Section. A measure of the reflecting 
ability of the target; it indicates the ratio of total power re- 
flected from the target to incident power density impinging 
from the radar. 

Effective Search Width. Twice the range of an equivalent 
definite range law of detection (equivalent, with respect to 
detecting a uniform distribution of targets). 

Effective Visibility, E. In parallel sweeping, half the sweep 
spacing for which the probability of detection is 0.5. 

Epoch. Any given instant of time. 

ERF. Error function or probability integral: 

erf x= ^ 

y ttJo 

Evasive Routing. Routing to avoid known submarine positions. 

Fix. Presumptive position of target as determined from a 
single observation. 

Fixation. The short stationary periods of actual seeing be- 
tween jumps of the eye in visual scanning motion. 

Fovea. Central part of retina, region of maximum acuity for 
daylight vision. 

Glimpse Probability, g. Probability that a target be sighted 
at a single glimpse. 

Homing Torpedo. Torpedo self-guided to its target by some 
property of the target. 

Known Random Distribution. A distribution in which the 
probabilities are known in advance. 

Lateral Range. The minimum distance (at closest approach) 
between target and observer. When the target is at rest, it is 
the perpendicular distance between the target and the ob- 
server’s track. 

Lateral Range Curve. A curve which gives the probability 
of detection of an object by an observer proceeding along a 
straight course as a function of distance of direct approach. 

Lead Angle. In this volume, the angle between the barrier 
line and line of flight of the observer aircraft. 

Limiting Approach Angle. Angle whose sine is the ratio of 
target velocity to observer velocity. This angle determines 
the slope of the lateral boundaries of the region of approach 
of target to observer. 


167 


168 


GLOSSARY 


Limiting Submerged Approach Angle. Limiting approach 
angle for a submerged submarine. 

Listening. Use of sonar to detect sonic and supersonic sounds 
generated by the target itself. 

Looking. Trying to detect with any of the means considered: 
visual, radar, sonar, etc. 

Maximum Sighting Range. The range at which the target 
contrast reaches the foveal threshold. 

One-Ping Probability. The probability of detection by using 
a single ping. 

Ping. Acoustic pulse signal projected by echo-ranging trans- 
ducer. 

PPI. Plan Position Indicator. 

Radar. Generic term applied to methods and apparatus that 
use radio for detection and ranging. 

Recognition Differential. The number of decibels by 
which a signal level must exceed the background level in 
order to be recognized 50 per cent of the time. 

Region of Approach. Area in which a craft, moving with 
velocity less than that of another craft, must have its start- 
ing point to be able to intercept the latter. 

Relative Bearing. Target bearing relative to own ship bear- 
ing. 

Relative Course, d. Angle between the observer’s vector 
velocity and the target’s relative velocity. 

Relative Velocity, w. The target’s vector velocity with 
respect to the observer. 

Reverberation. Sound scattered diffusely back toward the 
source, principally from the surface or bottom and from 
small scattering sources such as bubbles of air and suspended 
solid matter. 

Scanning Line. The locus of points on the ocean surface at 
which the detecting instrument is directed. 

ScHNORCHEL. U-Boat Combination air intake and exhaust tube 
that permits submerged diesel operation. 

Search Rate. See sweep rate. 

Search Width. See sweep width. 


Sea Return Area. Ocean surface area which reflects radar 
pulse back to the transmitter. 

Sonar. Generic term applied to methods and apparatus that 
use sound for navigation and ranging. 

Sweep Rate. A measure of the searching craft’s effectiveness 
in covering an area. It is the number of contacts made per 
hour per unit of target density by the searching craft, ex- 
pressed in square miles per hour. 

Sweep Spacing, S. Distance between track lines in parallel 
and equally spaced search. 

Sweep Width. Sweep rate divided by speed (approximately). 

Target. Object of search. 

Target Aspect. Orientation of the target as seen from own ship. 

Target Strength. A measure of the reflecting power of the 
target. Ratio in decibels of the target echo to the echo from 
a 6-ft diameter perfectly reflecting sphere at the same range 
and depth. 

Threshold Contrast. Just perceptible contrast of target 
against its background. 

Time of Fix. Time or epoch at which information about the 
point of fix is given. 

Torpedo Danger Zone. Area around a ship, and moving with 
the ship, within which a torpedo must be fired if it is to have 
any chance of scoring a hit. 

Track Angle, <f>. Angle between the vector velocities of ob- 
server and target. 

Track Problem. Selection of path and motion of the observer 
for assumed position and motion of the target. 

Transducer. Any device for converting energy from one form 
to another (electrical, mechanical, or acoustical). In sonar, 
usually combines the functions of a hydrophone and a pro- 
jector. 

m-Moving Space. Barrier flight plan as seen from a target 
moving with speed u. 

X-Band. Band of radar frequencies with wavelength approxi- 
mately 3 cm. 

Zigzag. Change course frequently to make attack by sub- 
marines difficult. 



CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS 


Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-20 

The Trustees of Columbia Uni- 
versity in the City of New York 

New York, New York 

Studies and experimental investigations in con- 
nection with and for the development of equip- 
ment and methods pertaining to submarine 
warfare. 

OEMsr-1128 

The Trustees of Columbia Uni- 
versity in the City of New York 

New York, New York 

Conduct studies and experimental investigations 
in connection with and for the development 
of equipment and methods involved in subma- 
rine and subsurface warfare. 






169 


SERVICE PROJECT NUMBERS 


The projects listed below were transmitted to the Office of the Execu- 
tive Secretary, OSRD, from the War or Navy Department through 
either the War Department Liaison Officer for NDRC or the Office of 
Research and Inventions (formerly the Coordinator of Research and 
Development), Navy Department. 


Service 

Project 

Number 


Title 


AC-50 


Operations Research 


NR-100 


Operations Research 



170 


INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 

For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


Aerial escort for ships, 137-165 
final sweeps, 162-165 
plan design, 160-162 
scouting effectiveness, 138-140 
screening effectiveness, 138, 140-142 
screening polar diagram, 142-160 
tactical situation, 137-138 
Air patrols against incoming subma- 
rines, 109-110 

Airborne microwave search radar, 62-64 
Aircraft sightings of submarines and 
surface ships, 57-61 

Apparent contrast, dependence on at- 
mospheric conditions, 52-53 
Approach region, 4-5, 132-133 
Area search 

parallel sweeps, 29-31, 92-94 
random search, 28-29 

Barrier patrols, 95-107 

advancing element barrier, 97-98, 
103-105 

back-and-forth patrol, 106 
construction of crossover patrol, 95- 
101 

definite range laws of detection, 

102- 104 

practical applications, 104-106 
retreating element barrier, 97, 98, 103 
stationary element barrier, 97, 98, 

103- 106 

target speed near observer speed, 

106- 107 

Barriers, circular 
see Circular barriers 
Bayes’ theorem, 37, 41, 112 
Blip-scan ratio, 67-70 
Brightness of sea, 53 

Circular barriers, 107-110 
air patrols against incoming subma- 
rines, 109-110 

constant radial flux of hostile craft, 

107- 108 

patrol against centrifugal targets, 110 
surface patrol against incoming sub- 
marines, 110 

targets moving toward central ob- 
jective, 108-109 

Contact probability in scanning, 49-51 
Convoy hit probability, single torpedo, 
120-128 

Cross-over barrier patrols 
see Barrier patrols 

Danger zone, 137 


Detection circle, 137 
Detection of target 
see Target detection 

Echo-ranging gear, standard, 85-92 
operational data, 91-92 
theoretical calculations, 85-91 
Effective search (sweep) width, 24-25 
Equations for search methods, 1-17 
motion at fixed speed and course, 1-4 
nonuniform distributions of targets, 
16-17 

random distributions of targets, 5-7 
random encounters with uniformly 
distributed targets, 7-15 
region of approach, 4-5 
ErfX (error function), definition, 28 
Expendable radio sono buoy, 82-85 
detection factors, 82 
operational data, 84-85 
Eye as detecting instrument 
see Visual detection 

Final sweeps by aerial escort, 162-165 
random submarines, 163-165 
submarines mounting guard, 164-165 
uniformly distributed, 163 
Force requirements for searching 
see Searching effort distribution 

Glimpse, 19 

Inverse cube law for plotting screening 
polar diagram 

see Plotting the screening polar dia- 
gram 

Inverse cube law of sighting, 22 

Lagrange multiplier, 39 
Lateral range distribution, 24-26 
Lateral range of target, 3-4 
Limiting approach angle and region, 4-5 
Linear scan, 50 

Maximum sighting range, 53-55 
Microwave search radar, airborne, 62-64 
Missiles, optimum destructiveness of, 
45-46 

Motion at fixed speed and course, 
equations, 1-4 

Optimum destructiveness of missiles, 
45-46 

Optimum scanning, 43-45 
Parallel sweeps (search), 29-31 


effective visibility, definition, 29-30 
in sonar search, 92-94 
sweep density, 31 
Patrolling stations, 135-136 
Pickets, 136 

Plan design for aerial escort, 160-162 
Plan position indicator (PPI), 63 
Plotting the screening polar diagram, 
146-155 

target moving on straight path, 152- 
155 

target on straight line interval, 148- 
152 

Potential, sighting, 23 
Probability function for torpedo hit, 
119-120 

Probability of contact in scanning, 
49-51 

Probability of convoy hit with single 
torpedo, 120-128 
contact-fuzed torpedo, 124 
curly torpedo, 127 
homing torpedo, 124-126 
straight run torpedo, 124-126 
Probability of detection, instantaneous, 
18-22 

Probability of hitting single task force - 
ship, 129-132 

Propagation of radar energy, 65-67 

Radar, airborne microwave search, 
62-64 

Radar detection, 62-74 
blip-scan ratio, 67-70 
comparison with visual search, 64-65 
computational methods, 72-74 
modern search radar characteristics, 
62-64 

range distributions and search width, 
70-72 

types of Naval targets, 68 
Radar propagation, 65-67 
Radar range distributions and search 
width, 70-72 

Radio sono buoy (expendable), 82-85 
Random encounters with uniformly 
distributed targets, 7-15 
Random search, 28-29 
Random target distributions, 5-7 
Relative speeds and velocities, 1-4 
Relative track, 2-3, 22 
Retiring search, 95 

Rules for obtaining screening polar 
diagram, 155-160 

Scanning, contact probability, 49-51 


171 


172 


INDEX 


area scan, 51 
linear scan, 50 
Scanning, optimum, 43-45 
Scouting effectiveness, aerial escort, 
138-140 

Screening effectiveness, aerial escort, 
138, 140-142 

Screening polar diagram, 142-160 
general considerations, 142-146 
instructions for obtaining diagram, 
155-160 

plotting (inverse cube law), 146-155 
Screens, sonar 
see Sonar screens 
Sea brightness, 53 
Search, random, 28-29 
Search about point of fix, 110-118 
retiring square search for moving 
target, 115-118 

square search for stationary target, 
111-114 

Search density, 38-39 
Search for targets in transit, 95-118 
barrier patrols, 95-107 
circular barriers, 107-110 
search about point of fix, 110-118 
Search methods, geometrical and sta- 
tistical, 1-17 

motion at fixed speed and course, 1-4 
nonuniform target distributions, 16-17 
random encounters with uniformly 
distributed targets, 7-15 
random target distributions, 5-7 
region of approach, geometric analy- 
sis, 4-5 

Search radar characteristics, modern, 
62-64 

Search width, effective, 24-25, 70-72 
Searching effort distribution, 35-46 
alternative search regions, 35-38 
distribution of effort in time, 42-43 
geometric construction, 40 
optimum destructiveness, 45-46 
optimum scanning, 43-45 
sample application, 41-42 
targets continuously distributed, 38-41 
Sighting potential, 23 
Sighting range, maximum, 53-55 
Sightings of submarines and surface 
ships from aircraft, 57-61 
Sonar background level, factors in- 
fluencing 


ambient noise in ocean, 78-79 
reverberation, 80 
self-noise, 79 
sonar characteristics, 80 
Sonar detection, 75-94 

background level, factors influencing, 
80 

expendable radio sono buoy, 82-85 
parallel sweeps, 92-94 
signal recognition, factors influencing, 
80-81 

signal strength, factors influencing, 76 
standard echo-ranging gear, 85-92 
Sonar screens, 119-136 

patrolling stations, 135-136 
pickets, 136 

placing of screen, 133-135 
probability of hitting task force ship, 
129-132 

single torpedo probability of convoy 
hit, 120-128 

submerged approach region, 132-133 
torpedo danger zones, hitting proba- 
bility, 119-120 

Sonar searching, parallel sweeps, 92-94 
Sonar signal recognition, factors in- 
fluencing 

background type, 81 
data presentation, 81 
operator skill, 81 
signal type, 80 

Sonar signal strength, factors influ- 
encing, 76-77 

intensity of transmitted pulse, 76 
receiver characteristics, 78 
reflecting power of target, 78 
sound output of ship, 76 
sound transmission, 77 
Sono buoy, expendable radio, 82-85 
detection factors, 82 
operational data, 84-85 
Square search (retiring) for a moving 
target, 115-118 

Square search for a stationary target, 
111-114 

Standard echo-ranging gear, 85-92 
Submerged approach region in the 
ocean, 132-133 

Surface patrol against incoming sub- 
marines, 110 

Sweep width, effective, 24-25, 70-72 
Sweeps, parallel, 29 


Sweeps by aerial escorts, 162-165 
random submarines, 163-165 
submarine mounting guard, 164-165 
uniformly distributed, 163-164 

Tactics with aerial escort, 137-138 
Target detection, 18-34 
basic facts of detection, 18 
dependence of track detection, 22-24 
distribution in true range, 26-28 
forestalling, 31-32 

instantaneous detection probabilities, 
18-22 

lateral range distribution, 24-26 
operational distribution, 32-34 
parallel sweeps, 29-31 
random search, 28-29 
Target’s relative track, 2-3, 22 
Targets, random distributions, 5-7 
Targets in transit, search, 95-118 
barrier patrols, 95-107 
circular barriers, 107-110 
search about point of fix, 110-118 
Threshold contrast, 48-49, 53 
Torpedo danger zones, 119-120 
“browning shot”, 120 
lethal coverage of weapon, 119-120 
Torpedo hit probability 
convoy, 120-128 
probability function, 119-120 
single task force ship, 129-132 
Trapping square, 95 

Visibility (meteorological), 52 
Visual detection, 47-61 
brightness contrast, 48 
contact probability and scanning, 
49-51 

dependence of apparent contrast on 
atmospheric conditions, 52-53 
eye construction, 47 
intrinsic sea brightness, 53 
maximum sighting range, 53-55 
operational situations analyzed, 57-61 
performance of eye, 47-49 
solid angle effect, 48-49 
target and background, 52-53 
target shape effect, 48-49 
threshold contrast, 48-49, 53 
visual perception angle, 55-57 

( 

Wake as sighting target, 54-55, 57-58 


REGRADED UNCLASSI^B- 
ORDEH SEC ARMY BY TAG PER [f 2 1 0 0 


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